Lectures in Probability Theory and Statistics 1995

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg E Takens, Groningen B. Teissier, Paris

1690

Springer

Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Singapore Tokyo

M. T. Barlow D. Nualart

Lectures on Probability Theory and Statistics Ecole d'Et6 de Probabilit6s de Saint-Flour X X V - 1995 Editor: P. Bernard

Springer

Authors

Editor

Martin T. Barlow Department of Mathematics University of British Columbia # 121-1984 Mathematics Road Vancouver, B.C. Canada V6T 1Z2

Pierre Bernard Laboratoire de Mathdmatiques Appliqudes UMR CNRS 6620 Universitd Blaise Pascal Clermont-Ferrand F-63177 Aubibre Cedex, France

David Nualart Department d'Estadistica Universitat de Barcelona Facultat de Matem~ttiques Gran Via de les Corts Catalanes, 585 E-08007 Barcelona, Spain Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufilahme L e c t u r e s on p r o b a b i l i t y t h c o r y a n d statistics / Ecole d'Et~ de Probabilitds de Saint-Flour X X V - 1995. M. T. Barlow ; D. Nualart. Ed.: P. Bernard. - Berlin ; Heidelberg ; N e w York ; Barcelona ; Budapest ; Hong Kong ; London ; Milan ; Paris ; Santa Clara ;

Singapore ; Tokyo : Springer, 1998 (Lecture notes in mathematics ; Vol. 1690) ISBN 3-540-64620-5 Mathematics Subject Classification (199 I): 60-01, 60-02, 60-06, 60D05, 60G57, 60H07, 60J15, 60J60, 60J65 ISSN 0075- 8434 ISBN 3-540-64620-5 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10649898 41/3143-543210 - Printed on acid-free paper

INTRODUCTION

This volume contains lectures givcn at the Saint-Flour Summer School of Probability Theory during the period 10th - 26th July, 1995. We thank the authors for all the hard work they accomplished. Their lectures are a work of refercncc in thcir domain. The school brought together i00 participants, 29 of whom gave a lecture concerning their research work. At the end of this volume you will find the list of participants and their papers. Finally, to facilitate research concerning previous schools we give here the number of the volume of "Lecture Notes" where they can be found :

L e c t u r e N o t e s in M a t h e m a t i c s 1971: n~ 1977:n~ 1982: n~ 1988: n~ 1993: n~

-

1973: n~ 1978 : n~ 1983:n~ 1989: n~ 1994: n~

L e c t u r e N o t e s in S t a t i s t i c s 1986 : n~

-

1974:n~ 1979: n~ 1984: n~ 1990: n~ 1996:n~

-

1975 : 1980: 19851991:

n~ n~ 1986 et n~

1976:n~ 1981:n~ 1987: n~ 1992:n~

-

TABLE OF CONTENTS

Martin 1

T. BARLOW

Introduction

: "DIFFUSIONS

ON FRACTALS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

2

T h e Sierpinski Gasket . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

3

Fractional Diffusions

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

4

Dirichlet Forms, Markov P r o c e s s e s and Electrical Networks . . . . . . . . . .

46

5

G c o m e t r y of R e g u l a r Finitely Ramified Fractals

. . . . . . . . . . . . . . .

59

6

R e n o r m a l i z a t i o n on Finitely Ramified Fractals

. . . . . . . . . . . . . . . .

79

7

Diffusions on p.c.f.s.s, sets

. . . . . . . . . . . . . . . . . . . . . . . . . .

94

8

T r a n s i t i o n Density E s t i m a t e s References

David

. . . . . . . . . . . . . . . . . . . . . . . . .

106

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114

NUALART

: "ANALYSIS ON WIENER SPACE AND ANTICIPATING STOCHASTIC CALCULUS"

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

125

Derivative a n d divergence o p e r a t o r s on a G a u s s i a n space . . . . . . . . . . .

126

1.1 Derivative o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

126

1.2 Divergence o p e r a t o r . . . . . . . . . . . . . . . . . . . . . . . . . . . .

130

1.3 Local p r o p e r t i e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

131

1.4 W i e n e r chaos e x p a n s i o n s

. . . . . . . . . . . . . . . . . . . . . . . . .

133

. . . . . . . . . . . . . . . . . . . . . . . . . . .

135

1.5 T h e w h i t e noise case

1.6 S t o c h a s t i c integral r e p r e s e n t a t i o n of r a n d o m variables 2

123

. . . . . . . . . .

139

O r n s t e i n - U h l e n b e c k semigroup a n d equivalence of n o r m s . . . . . . . . . . .

141

2.1 M e h l e r ' s f o r m u l a

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

141

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

2.2 H y p e r c o n t r a c t i v i t y

2.3 G e n e r a t o r of t h e O r n s t e i n - U h l e n b e c k s e m i g r o u p

. . . . . . . . . . . . .

145

2.4 M e y e r ' s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . .

146

VLLI

3

A p p l i c a t i o n of Malliavin calculus to s t u d y p r o b a b i l i t y laws . . . . . . . . . .

155

3.1 C o m p u t a t i o n of p r o b a b i l i t y densities

155

. . . . . . . . . . . . . . . . . . .

3.2 R e g u l a r i t y of densities a n d c o m p o s i t i o n of t e m p e r e d d i s t r i b u t i o n s w i t h e l e m e n t s of ~)-oo

. . . . . . . . . . . . . . . . . . . . . . . . . .

3.3 T h e case of diffusion processes . . . . . . . . . . . . . . . . . . . . . . 3.4 Lp e s t i m a t e s of t h e density a n d a p p l i c a t i o n s . . . . . . . . . . . . . . . 4

5

160 163 164

Support theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

4.1 P r o p e r t i e s of t h e s u p p o r t

174

. . . . . . . . . . . . . . . . . . . . . . . . .

4.2 S t r i c t p o s i t i v i t y of t h e density a n d skeleton . . . . . . . . . . . . . . . .

177

4.3 Skeleton a n d s u p p o r t for diffusion processes

182

. . . . . . . . . . . . . . .

4.4 V a r a d h a n e s t i m a t e s . . . . . . . . . . . . . . . . . . . . . . . . . . . .

183

A n t i c i p a t i n g s t o c h a s t i c calculus . . . . . . . . . . . . . . . . . . . . . . . .

188

5.1 S k o r o h o d i n t e g r a l processes . . . . . . . . . . . . . . . . . . . . . . . .

188

5.2 E x t e n d e d S t r a t o n o v i c h intcgral . . . . . . . . . . . . . . . . . . . . . .

197

5.3 S u b s t i t u t i o n f o r m u l a s . . . . . . . . . . . . . . . . . . . . . . . . . . .

201

A n t i c i p a t i n g s t o c h a s t i c differential e q u a t i o n s

210

. . . . . . . . . . . . . . . . .

6,1 S t o c h a s t i c differential e q u a t i o n s in t h e S t r a t o n o v i c h sense 6.2 S t o c h a s t i c differential e q u a t i o n s w i t h b o u n d a r y c o n d i t i o n s

........ ........

210 215

6.3 S t o c h a s t i c differential e q u a t i o n s in t h e S k o r o h o d sense . . . . . . . . . .

217

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

221

DIFFUSIONS

ON FRACTALS

Martin T. BARLOW

1. I n t r o d u c t i o n . T h e notes are based on lectures given in St. Flour in 1995, and cover, in greater detail, most of the course given there. T h e word "fractal" was coined by M a n d e l b r o t [Man] in the 1970s, b u t of course sets of this t y p e have been familiar for a long time - their early history being as a collection of pathological examples in analysis. There is no generally agreed exact definition of the word "fractal", and a t t e m p t s so far to give a precise definition have been unsatisfactory, leading to classes of sets which are either too large, or too small, or both. This a m b i g u i t y is not a p r o b l e m for this course: a more precise title would be "Diffusions on some classes of regular self-similar sets". Initial interest in the properties of processes on fractals came from m a t h e m a t i c a l physicists working in t h e t h e o r y of disordered media. Certain m e d i a can be modelled by p e r c o l a t i o n clusters at criticality, which are expected to exhibit fractal-like properties. Following t h e initial p a p e r s [AO], [RT], [GAM1-GAM3] a very substantial physics l i t e r a t u r e has developed - see [HBA] for a survey and bibliography. Let G be an infinite s u b g r a p h of Z a. A simple r a n d o m walk (SRW) (X,~, n > 0) on G is j u s t the Markov chain which moves from x E G with equal p r o b a b i l i t y to each of the neighbours of x. Write p n ( x , y) = IP~(X,~ = y) for the n-step t r a n s i t i o n probabilities. If G is the whole of Z d then E(X,~) 2 = n with m a n y familiar consequences - the process moves roughly a distance of order v/n in time n, and the p r o b a b i l i t y law p,~(x, .) p u t s most of its mass on a ball of radius cdn. If G is not the whole of ~d then the movement of the process is on the average r e s t r i c t e d by the removal of p a r t s of the space. Probabilistically this is not obvious b u t see [DS] for an elegant argument, using electrical resistance, t h a t the removal of p a r t of the s t a t e space can only make the process X 'more recurrent'. So it is not unreasonable to expect t h a t for certain graphs G one m a y find t h a t the process X is sufficiently r e s t r i c t e d t h a t for some fl > 2 -

(1.1)

E ~ (X~ - x) ~ • n2/~.

(Here and elsewhere I use • to m e a n ' b o u n d e d above and below by positive constants', so t h a t (1.1) means t h a t there exist constants el, c2 such t h a t c l n 2/~ < E ~ ( X , - x) ~ < c2n~/a). In [AO] and [RT] it was shown t h a t if V is the Sierpinski gasket (or more precisely an infinite graph based on the Sierpinski gasket - see Fig. 1.1) t h e n (1.1) holds w i t h / 3 = log 5 / l o g 2.

Figure 1.1: T h e graphical Sierpinski gasket.

Physicists call b e h a v i o u r of this kind by a r a n d o m walk (or a diffusion - t h e y are not very i n t e r e s t e d in the distinction) subdiffusive - the process moves on average slower t h a n a s t a n d a r d r a n d o m walk on Z d. Kesten [Ke] proved t h a t the SRW on the 'incipient infinite cluster' C (a percolation cluster at p = Pc b u t conditioned to be infinite) is subdiffusive. T h e large scale structure of C is given by taking one infinite p a t h (the ' b a c k b o n e ' ) together with a collection of 'dangling ends', some of which are very large. K e s t e n a t t r i b u t e s the subdiffusive behaviour of SRW on C to the fact t h a t the process X spends a substantial amount of time in the dangling ends. However a g r a p h such as the Sierpinski gasket (SG) has no dangling ends, and one is forced to search for a different explanation for the subdiffusivity. This can be found in t e r m s of t h e existence of 'obstacles at all length scales'. W h i l s t this holds for the graphical Sierpinski gasket, the n o t a t i o n will be slightly simpler if we consider a n o t h e r example, the graphical Sierpinski c a r p e t (GSC). (Figure 1.2).

-t : H - H - H : H - H q

i i

,;fi fi:H:H: ; f:H-t [IIIII

q

t: : H F

. . . . . . . . .

f:H-H:H:H-H:

II

;H:F :H-F " I I I I I

III

Figure 1.2: T h e graphical Sierpinski carpet. This set can be defined precisely in the following fashion. Let H0 -- Z ~. For o~ x = (n, m) e H0 write n,m in t e r n a r y - so n -- ~i=o ni3", where ni 9 {0, 1,2}, and n~ -- 0 for all b u t finitely m a n y i. Set J~ -- { ( m , n ) : nk -- 1 and mk -- 1}, so t h a t Jk consists of a union of disjoint squares of side 3k: the square in Jk closest to t h e origin is { 3 ~ , . . . , 2 . 3 ~ - 1} • { 3 k , . . . , 2 . 3 k - 1}. Now set (1.2)

H,, = Ho- O Jk, k:l

H= 5 H,~. n=O

:~_

11111

IIII

IIII]

IIlI

-H:t

lIII

ilIi

itti

IIII iIII

IIlI

~HIIII

IlII

!

IIII

i

!~H~

IlIl

!HI::

Figure 1.3: T h e set H1 9

14' ; ~-

'.

.

.

.

!

~

!

T

F!

---

i

Li

i

i

i

Fi

.__

i

!i

i

i

i

i

i

ttt

r

ii

-L

!E:_I

t

i

1"

__.

---

!ii_I

H --~F-H

4~f-H--t:t--

.

Figure 1.4: T h e set/-/2 9 Note t h a t H A [0, 3n] 2 = Hn N [0, 3n] 2, so t h a t the difference between H and Hn will only be d e t e c t e d by a SRW after it has moved a distance of 3 n from the origin. Now let X (n) be a SRW on Hn, s t a r t e d at the origin, and let X be a SRW on H. T h e process X (~ is j u s t SRW on Z~_ and so we have (1.3)

E(X(~ 2 ~" n.

T h e process XO) is a r a n d o m walk on a the intersection of a t r a n s l a t i o n invariant subset of 7..2 w i t h 7.~_. So we expect 'homogenization': the processes n - 1 / 2 X [,~], O) t _> 0 should converge weakly to a constant multiple of Brownian motion in ~ 2 . So, for large n we should have E ( X O ) ) 2 ~ a l n , and we would expect t h a t al < 1, since the obstacles will on average t e n d to i m p e d e the motion of the process. Similar considerations suggest t h a t , writing ~on(t) = 1EO(X}n)) 2, we should have ~pn(t) " a n t

as t --* oo.

However, for small t we would expect t h a t ~n and ~0,~+1 should be a p p r o x i m a t e l y equal, since the process will not have moved far enough to detect the difference between Hn and H n + l . More precisely, if tn is such t h a t ~On(tn) = (3n) 2 then ~on

and ~o=+1 should b e a p p r o x i m a t e l y equal on [O, tn-bl ]. So we m a y guess t h a t the b e h a v i o u r of the family of functions ~o,~(t) should be roughly as follows: (1.4)

~on(t) = bn + an(t - in),

v.+l(s) = v.(s),

t >_ t,~,

0 < s < t~+l.

If we a d d the guess t h a t an = 3 - ~ for some ~ > 0 then solving the equations above we deduce t h a t tn ~ 3 (2+a)'~, bn ~- 3 2n. So ff ~o(t) = E ~ 2 t h e n as 4o(t) ~- lira= ~on(t) we deduce t h a t ~v is close to a piecewise linear function, and t h a t ~(t) • t 2/z where fl = 2 + a. T h u s the r a n d o m walk X on the graph H should satisfy (1.1) for some f~ > 2. T h e a r g u m e n t given here is not of course rigorous, b u t (1.1) does a c t u a l l y hold for t h e set H - see [BB6, BBT]. (See also [Jo] for the case of the graphical Sierpinski gasket. T h e proofs however run along r a t h e r different lines t h a n the heuristic a r g u m e n t sketched above). Given b e h a v i o u r of this t y p e it is n a t u r a l to ask if the r a n d o m walk X on H has a scaling limit. More precisely, does there exist a sequence of constants Tn such t h a t t h e processes (1.5)

( 3 - ' ~ X i t / , . ] , t > 0)

converge weakly to a non-degenerate limit as n --+ co? For the graphical Sierpinski c a r p e t the convergence is not known, though there exist m= such t h a t the family (1.5) is tight. However, for the graphical Sierpinski gasket the answer is 'yes'. Thus, for certain very regular fractal sets F C I~d we are able to define a limiting diffusion process X = ( X t , t > O, P~, x C F ) w h e r e ]?~ is for each x E F a p r o b a b i l i t y measure on fl = {w Z C([0, co), F ) : w(0) = x}. Writing T t S ( x ) = E ~ f ( X t ) for the semigroup of X we can define a 'differential' o p e r a t o r s defined on a class of functions 7:)(/:F) C C ( F ) . In m a n y cases it is reasonable to call s the Laplacian on F.

F r o m t h e process X one is able to o b t a i n information a b o u t the solutions to the Laplace a n d h e a t equations associated with L:F, the heat equation for example t a k i n g the form Ou

(1.6)

0-7 = LFu, u(0, x) = u0(x),

where u = u ( t , x ) , x E F , t > O. The wave equation is r a t h e r harder, since it is not very susceptible to probabilistic analysis. See, however [KZ2] for work on the wave equation on a some manifolds with a 'large scale fractal structure'.

T h e m a t h e m a t i c a l l i t e r a t u r e on diffusions on fractals and their associated infinitesimal generators can be divided into b r o a d l y three parts: 1. Diffusions on finitely ramified fractals. 2. Diffusions on generalized Sierpinski carpets, a family of infinitely ramified fractals. 3. S p e c t r a l p r o p e r t i e s of the ' L a p l a c i a n ' s These notes only deal with the first of these topics. On the whole, infinitely ramified fractals are significantly h a r d e r t h a n finitely ramified ones, and sometimes require a very different approach. See [Bas] for a recent survey. These notes also contain very little on spectral questions. For finitely ramified fractais a direct a p p r o a c h (see for example [FS1, Shl-Sh4, KL]), is simpler, and gives more precise information t h a n the heat kernel m e t h o d based on estimating

/F

p ( t , x , x ) d x = y ~ e-~' ~. i

In this course Section 2 introduces the simplest case, the Sierpinski gasket. In Section 3 I define a class of well-behaved diffusions on metric spaces, "Fractional Diffusions", which is wide enough to include m a n y of the processes discussed in this course. It is possible to develop their properties in a fairly general fashion, without using much of the special structure of the state space. Section 4 contains a brief i n t r o d u c t i o n to t h e theory of Dirichlet forms, a n d also its connection with electrical resistances. T h e remaining chapters, 5 to 8, give the construction and some p r o p e r t i e s of diffusions on a class of finitely ramified regular fractals. In this I have largely followed the analytic ' J a p a n e s e ' approach, developed by Kusuoka, Kigami, ~'klkushima a n d others. Many things can now be done more simply t h a n in the early probabilistic work - b u t there is loss as well as gain in a d d e d generality, and it is worth pointing out t h a t the early papers on the Sierpinski gasket ([Kusl, Go, BP]) contain a wealth of interesting direct calculations, which are not r e p r o d u c e d in these notes. A n y reader who is surprised by the a b r u p t end of these notes in Section 8 should recall t h a t some, at least, of the properties of these processes have a l r e a d y been o b t a i n e d in Section 3. c~ denotes a positive real constant whose value is fixed within each L e m m a , T h e o r e m etc. Occasionally it will b e necessary to use n o t a t i o n such as c3.5.4 - this is s i m p l y the constant c4 in Definition 3.5. c, c r, c ~ denote positive real constants whose values m a y change on each appearance. B(x, r) denotes the open ball with centre x a n d radius r, and if X is a process on a metric space F then

TA = inf{t > O : Xt E A } , Ty=inf{t>0:Xt=y}, ~-(x, r) = inf{t > 0 : Xt fd B(x, r)}. I have included in the references most of the m a t h e m a t i c a l papers in this area known to me, and so t h e y contain m a n y papers not mentioned in the text. I a m grateful to G e r a r d Ben Arous for a n u m b e r of interesting conversations on the physical conditions under which subdiffusive behaviour might arise, to Ben H a m b l y

for checking the final manuscript, and to Ann Artuso and Liz ttowley for their typing.

Acknowledgements. This research is supported by a NSERC (Canada) research grant, by a grant from the Killam Foundation, and by a EPSRC (UK) Visiting Fellowship.

2. T h e Sierpinski Gasket This is the simplest non-trivial connected symmetric fractal. The set was first defined by Sierpinski [Siel], as an example of a pathological curve; the name "Sierpinski gasket" is due to Mandelbrot [Man, p.142]. Let Go = {(0, 0), (1, 0), (1/2, ~/3/2)} = {a0, al, a2} be the vertices of the unit triangle in I~2, and let Tlu(Go) = Ho be the closed convex hull of Go. The construction of the Sierpinski gasket (SG for short) G is by the following Cantor-type subtraction procedure. Let b0, bl, b2 be the midpoints of the 3 sides of Go, and let A be the interior of the triangle with vertices {bo,bl,b2}. L e t / / 1 = H0 - A, so that HI consists of 3 closed upward facing triangles, each of side 2 -1. Now repeat the operation on each of these triangles to obtain a set H2, consisting of 9 upward facing triangles, each of side 2 -2 .

Figure 2.1: The sets//1 and H2. Continuing in this fashion, we obtain a decreasing sequence of closed non-empty ~o and set sets ( H n)==0, (2.1)

G = 5 n=0

/In.

Figure 2.2: The set H4. It is easy to see t h a t G is connected: just note that OHn C H,n for all m > n, so t h a t no point on the edge of a triangle is ever removed. Since IHnl = (3/4)nlH0 I, we clearly have t h a t IGI = 0. We begin by exploring some geometrical properties of G. Call an n-triangle a set of the form G N B, where B is one of the 3 '~ triangles of side 2 -'~ which make up Hn. Let #n be Lebesgue measure restricted to H,~, and normalized so that # n ( H n ) -- 1; that is #,~(dx) = 2- (4/3)'~1/./~ (x) dz. Let /~G = wlim#,~; this is the n a t u r a l "flat" measure on G. Note that # c is the u n i q u e measure on G which assigns inass 3 -'~ to each n-triangle. Set df=log3/log2~ 1.58... L e m m a 2.1. For x E G, O < r < l

3-1r d' 0. Since B(x, r) can intersect at most 6 n-triangles, it follows that

(B(x, r)) < 6.3

:

= 18(2-("+U) d, < 18r d'. As each (n + 1)-triangle has diameter 2 - ( n + U , B ( x , r ) must contain at least one (n + 1)-triangle a n d therefore

#r

> 3 -('~+1) -- 3-1(2-'~) d' > 3-1r d'.

[]

Of course the constants 3 -1 , 18 in (2.2) are not important; what is significant is t h a t the #G-mass of balls in G grow as r ds. Using terminology from the geometry of manifolds, we can say that G has volume growth given by r d~ .

Detour on Dimension. Let (F, p) be a metric space. There are a number of different definitions of dimension for F and subsets of F : here I just mention a few. The simplest of these is box-counting dimension. For e > 0, A C F , let N(A,e) be the smallest number of balls B(x, e) required to cover A. Then

(2.3)

d i m B c ( A ) = lim sup ~0

log N(A, e) log e-x

To see how this behaves, consider some examples. We take (F, p) to be ~a with the Euclidean metric.

Examples.

1. Let A -- [0,1] a C ~a. Then

N(A, e)

x e -a, and it is easy to verify

that lira log N([0, 1]d, e) = d. ~0 log e - 1 2. The Sierpinski gasket G. Since G C H~, and H,, is covered by 3 ~ triangles of side 2 - n , we have, after some calculations similar to those in Lemma 2.1, that N ( G , r ) y- (1/r)l~176 So,

dimBc(G) = 3. Let A = Q N [0,1]. Then N(A,e) dimBe({p}) = 0 for any p E A.

log3 log 2"

• ~-1, so dimBc(A)

= 1. On the other hand

We see t h a t box-counting gives reasonable answers in the first two cases, but a less useful number in the third. A more delicate, but more useful, definition is obtained if we allow the sizes of the covering balls to vary. This gives us Hausdorff dimension. I will only sketch some properties of this here - for more detail see for example the books by Falconer [Fal, Fa2]. Let h : ]~+ --* R+ be continuous, increasing, with h(0) = 0. For U C F write diam(U) -- sup{p(~, y) : z, y E U} for the diameter of V. For 6 > 0 let

~(a)--inf{E h(d(U,))

: A c gu,,

i

diam(U,) < 6}.

i

Clearly 7-/h(A) is decreasing in 6. Now let (2.4) we call T/h(-)

7/h(A) = lim 7/~(A); 610

Hausdorff h-measure. Let

B ( F ) be the Borel a-field of F.

10 Lemma

2.2. 7-/h is a measure on (F, B(F)).

For a proof see [Fal, Chapter 1]. We will be concerned only with the case h(x) = x~: we then write ~ for 7~h. Note that a ~ 7-/~(A) is decreasing; in fact it is not hard to see that 7 ~ ( A ) is either +o0 or 0 for all but at most one a. Definition

2.3. The Hausdorff dimension of A is defined by dimH(A) = inf{~ : ~ ( A )

Lemma

= 0} -- s u p { a : 7-/~(A) = + ~ } .

2.4. dimH(A) < dimBc(A).

Proof. Let a > dimBc(A). Then as A can be covered by N(A, e) sets of diameter 2e, we have 7-/~(A) < N(A,E)(2e) ~ whenever 2e < 6. Choose 0 so that d i m B c ( A ) < a - 0 < a; then (2.3) implies that for all sufficiently smMl e, N ( A , e) _< e-(~-0). So ? ~ ( A ) : 0, and thus 7/~(A) : 0, which implies that dimH(A) < a. [] Consider the set A = Q N [0, 1], and let A -- {Pa,P2,..-} be an enumeration of A. Let 5 > 0, and Ui be an open internal of length 2 - i A 6 containing Pi. Then (Ui) covers A, so that 7-/~(A) < ~ i ~ 1 (6 A 2-i) ~, and thus 7 ~ ( A ) -- 0. So dimH(A) = 0. We see therefore that dimH can be strictly smaller than dimBc, and that (in this case at least) dimH gives a more satisfactory measure of the size of A. For the other two examples considered above Lemma 2.4 gives the upper bounds dimH([0, 1] d) < d, dimH(G) < log 3 / l o g 2. In both cases equality holds, but a direct proof of this (which is possible) encounters the difficulty that to obtain a lower bound on 7 ~ ( A ) we need to consider all possible covers of A by sets of diameter less than 6. It is much easier to use a kind of dual approach using measures. 2.5. Let # be a measure on A such that #( A ) > 0 and there exist Cl < c~, ro > O, such that

Theorem

(2.5)

p ( B ( x , r ) ) < e a r ~,

xeA,

r (v/3/2)2 - ~

for a E A, b E B.

Let x, y E G a n d choose n so t h a t

(v /2)2 -("+a) < I x - yl
0) such t h a t

(2.7)

(y(n) [~.~1, t

_> 0)

(xt,~ _> 0).

We have two problems: (1) How do we find the right ( a n ) ? (2) How do we prove convergence? We need some more notation. D e f i n i t i o n 2.13. Let ,,q,~ be the collection of sets of the form G N A, where A is an n-triangle. We call the elements of ,S,~ n-complexes. For x C Gn let Dn(x) = [.J{S c

S~:zcS}. T h e key p r o p e r t i e s of the SG which we use are, first t h a t it is very symmetric, a n d secondly, t h a t it is finitely ramified. (In general, a set A in a metric space F is

14

finitely ramified if there exists a finite set B such t h a t A - B is not connected). For the SG, we see t h a t each n-complex A is disconnected from the rest of the set if we remove the set of its corners, t h a t is A f3 Gn. T h e following is the key observation. Suppose Y0( ' ) = y E G,~-I (take y r Go for simplicity), a n d let T = inf{k > 0 : y(n) e Gn-1 - {y}}. Then y ( n ) can only escape from D~-I(y) at one of the 4 points, { x l , . . . , x4} say, which are neighbours of y in the graph ( G n _ l , E n - ~ ) . Therefore Y('~) E { X l , . . . , x4 }. F u r t h e r the s y m m e t r y of the set G= (3 D=(y) means t h a t each of the events {YT(n) = xi} is equally likely. X2

X3

Xl

Y

X4

Figure 2.4: y and its neighbours. Thus

(Y(T~)=x~

y ( n ) = y ) 1= ~ ,

a n d this is also equal to ~(Y1(~-1) -- xiIY(~-1) = y). (Exactly the same argument applies if y C Go, except t h a t we t h e n have only 2 neighbours instead of 4). It follows t h a t Y(~) looked at at its visits to G ~ - I behaves exactly like y ( , ~ - l ) . To state this precisely, we first make a general definition. D e f i n i t i o n 2.14. Let T = ]~+ or Z+, let (Zt,t E 7~) be a cadlag process on a metric space F , a n d let A C F be a discrete set. T h e n successive disjoint hits by Z on A are t h e s t o p p i n g times To, T 1 , . . . defined by T0=inf{t>0:Zt

(2.8)

EA},

T~+I = i n f { t > T ~ : Z t c A - { Z T ~ } } ,

n>_O.

W i t h this notation, we can summarize the observations above. Lemma

2.15. Let (Ti)i>_o be successive disjoint hits by y(,O on G~-I.

Then

(YT(?), i _> O) is a simple random wMk on G,~-I and is therefore equal in law to

(y(,-1), i _> 0). Using this, it is clear t h a t we can build a sequence of "nested" r a n d o m walks on G,~. Let N _> 0, and let y(N),~ k _> 0 be a SRW on GN with y(N) = 0. Let

[TN,m~ 0 _o be successive disjoint hits by y(N) on Gin, and set

y(m) = y ( m ( T N , m ) = y (TuN,) m '

i > 0.

15 It follows from L e m m a 2.15 t h a t y(m) is a SRW on Gin, and for each 0 < n < m 0 a.s. takes a little more work. (See [Har, p. 15]). In addition, if

~o(u) = Be - ' w then ~o satisfies the functional equation (2.11)

~o(5u) = f(~oCu)),

~J(O) ----- 1 .

We have a similar result in general. Proposition

2.17. Fix m > O. The processes

r~m , Z(i) = T~'m - Ti_I

n > m

are branching processes with offspring distribution T, and Z (i) are independent. Thus there exist W(i m) such that for each m (W(im),i >_ O) are independent, 5-row, and n,m ) --+ W.(trn) a.$. 5--n ( T ? ' m - Ti_l Note in particular that E(T~ '~ ) ---- 5 n, that is that the mean time taken by y ( , 0 to cross G~ is 5 ~. In terms of the graph distance on G~ we have therefore that y(n) requires roughly 5 '~ steps to move a distance 2n; this may be compared with the corresponding result for a simple random walk on Z d, which requires roughly 4 n steps to move a distance 2 n. The slower movement of Y(n) is not surprising - - to leave Gn f) B(O, 1/2), for example, it has to find one of the two 'gateways' (1/2, 0) or (1/4, vf3/4). Thus the movement of y(n) is impeded by a succession of obstacles of different sizes, which act to slow down its diffusion.

17 Given the space-time scaling of y ( , 0 it is no surprise that we should take a s = 5 ~ in (2.7). Define X~'

= v(n)

"[5-t],

t >_ O.

I n view of the fact that we have built the y(n) with the nesting property, we can replace the weak convergence of (2.7) with a.s. convergence. T h e o r e m 2.18. The processes X '~ converge a.s., and uniformly on compact interwa/s, to a process Xt, t >__O. X is continuous, a n d X t E G for all t > O.

Proof. For simplicity we will use the fact that W has a non-atomic distribution function. Fix for now rn > 0. Let t > 0. Then, a.s., there exists i = i(w) such that i

i+1

Ew?

Ew?).

j=l

j----1

a s w j~) = l i m . _ ~ 5-" (T? '~ - T?:7) it follows that for n > n0(o~), n~m z ? '~ < 5'~t < Ti+x 9

(2.12)

Now Y ( - ) ( T ? ,~) = r, ('~) by (2.9). Since r~('~) ~ D . ( r , (~)) for T? '~ < k < T3;1, we have [l~:~]-- - Y/(m) I < 2 - ~

for all n >__no.

This implies that [X~ - X~' I < 2 -'~+1 for n, n ' > no, so that X~ is Cauchy, and converges to a r.v. Xt. Since X ~ 6 Gn, we have X t 6 G. W i t h a little extra work, one can prove that the convergence is uniform in t, on compact time intervals. I give here a sketch of the argument. Let a 6 N, a n d let

~m :

rain l 0 a.s. Choose no such that for n > no i

5-nTt'm-

E

W (m) < l~n ,

l no. This implies

t h a t if T,~ = ~ i = as~ 1 W (m), and S < Tin, then sup I X : O 0, ?~, x 9 G) we need to construct X at arbitrary starting points x 9 G, and to show that (in some appropriate sense) the processes started at close together points x and y are close. This can be done using the construction given above - - see [BP, Section 2]. However, the argument, although not really hard, is also not that simple. In the remainder of this section, I will describe some basic properties of the process X , for the most part without giving detailed proofs. Most of these theorems will follow from more general results given later in these notes. Although G is highly symmetric, the group of global isometries of G is quite small. We need to consider maps restricted to subsets. D e f i n i t i o n 2.19. Let (F, p) be a metric space. A local isometry of F is a triple (A, B, ~o) where A, B are subsets of F and ~ois an isometry (i.e. bijective and distance preserving) between A and B, and between OA and OB. Let ( X t , t > O,g~,x 9 F ) b e a M a r k o v process on F. For H C F , set TH = inf{t > 0 : X t 9 H } . X is invariant with respect to a local isometry (A, B, ~o) if

P= ( ~ ( x , ^ r o ~ ) 9

t > 0) = P~(=)

(x,^ro, 9 t > o).

X is locally isotropic if X is invariant with respect to the local isometries of F. T h e o r e m 2.20. (a) There exists a continuous strong Markov process X : (Xt, t > O, ? L x 9 G) on G.

(b) The semigroup on C(G) de~ed by Pal(x) = E= y ( x , ) is Fe//er, and is pG-symmetric:

/G f(x)P~g(x)~G(d~) = /

a(~)Pd(x)~G(dx).

19

(c) X is locally isotropic on the spaces ( a , I" - " I) and ( G, da). (d) For n > 0 let Tn,i, i > 0 be successive disjoint hits by X on Gn. Then ~('*) = ~(~) XT..~, i > 0 defines a S R W on Gn, and =[5*t] -+ X t uniformly on compacts, a.s. So, in particular (Xt, t > 0, I?~ is the process constructed in Theorem 2.18. This theorem will follow from our general results in Sections 6 and 7; a direct proof may be found in [BP, Sect. 21. The main labour is in proving (a); given this (b), (c), (d) all follow in a relatively straightforward fashion from the corresponding properties of the approximating random walks ~(n). The property of local isotropy on (G, da) characterizes X : T h e o r e m 2.21. (Uniqueness). Let (Zt, t > O, Qz, x C ~) be a non-constant locally isotropic diffusion on (G, da). Then there exists a > 0 such that (F(z,

~ .,t ___0) = ? = ( x o , ~ .,t _> 0).

(So Z is equal in law to a deterministic time change of X ) . The beginning of the proof of Theorem 2.21 runs roughly along the lines one would expect: for n > 0 let (~(=), i >_ 0) be Z sampled at its successive disjoint hits on Gn. The local isotropy of ~z implies that Y(=) is a SRW on G=. However some work (see [BP, Sect. 8]) is required to prove that the process Y does not have traps, i.e. points z such that Q* (Yt = z for all t) = 1. R e m a r k 2.22. The definition of invariance with respect to local isometries needs some care. Note the following examples. 1. Let x , y E Gn be such that D , ( z ) n Go = ao, Dn(y) C3Go = 0. Then while there exists an isometry ~ from D,~(z) n G to D=(y) n G, ~v does not map OnD=(z) ;3 G to ORDn(y) n G. (OR denotes here the relative boundary in the set G). 2. Recall the definition of Hn, the n-th stage in the construction of G, and let B , = OH,~. We have G = cl(UB,). Consider the process Zt on G, whose local motion is as follows. If Zt E H,~ - H=-I, then Zt runs like a standard 1-dimensional Brownian motion on H , , until it hits Hn-1. After this it repeats the same procedure on Hn-1 (or H,~-k if it has also hit H,,-k at that time). This process is also invariant with respect to local isometries (A, B, ~o) of the metric space (G, I" - " I). See [He] for more on this and similar processes. To discuss scale invariant properties of the process X it is useful to extend G to an unbounded set G with the same structure. Set = U 2'~G, rt=0

and let G~ be the set of vertices of n-triangles in G,~, for n > 0. We have G~ = U 2kGn+k, k=0

20 and if we define Gm = {0} for m < 0, this definition also makes sense for n < 0. We can, almost exactly as above, define a limiting diffusion .~ = (_~,, t _> 0, ~x, x E G) Oil G: =

t _> o, a s

where (Y~('~),n > 0, k _> 0) are a sequence of nested simple random walks on G=, and the convergence is uniform on compact time intervals. The process X satisfies an analogous result to Theorem 2.20, and in addition satisfies the scaling relation (2.13)

F~(22~ 9 -, t > 0) : ~2~(25t 9 -, t > 0).

Note that (2.13) implies that _~ moves a distance of roughly t l~176 Set d,~ = d~(G) -- l o g 5 / l o g 2 .

in time t.

We now turn to the question: "What does this process look like?" The construction of X, and Theorem 2.20(d), tells us that the 'crossing time' of a 0-triangle is equal in law to the limiting random variable W of a branching process with offspring p.g.f, given by f ( s ) = s 2 / ( 4 - 3 s ) . From the functional equation (2.11) we can extract information about the behaviour of ~o(u) = E e x p ( - u W ) as u --* oo, and from this (by a suitable Tauberian theorem) we obtain bounds on P ( W < t) for small t. These translate into bounds on Px(1Xt - x I > A) for large A. (One uses scaling and the fact that to move a distance in G greater than 2, X has to cross at least one 0-triangle). These bounds give us many properties of X. However, rather than following the development in [BP], it seems clearer to first present the more delicate bounds on the transition densities of .Y and X obtained there, and derive all the properties of the p r o c ~ s from them. Write ~ c for the analogue o f # a for G, and Pt for the semigroup of X. Let ~. be the infinitesimal generator of Pt. 2.23. fit and Pt have densities ~(t, x, y) and p(t, x, y) respectively. ~(t, ~, y) i~ continuous on (0, ~ ) • ~ • ~. ~(t,~,y) = ~(t,y,~) for ~ t,~,y. t -~ ~ ( t , . , y) i~ C ~ on (0, ~ ) for each (~, y). For each t, y

Theorem

(a) (b) (c) (d)

I~(t,x,y) - ~(t,x',y)] - ~-~2-('~+1)

if y C G N D , ~ ( x ) c .

T h e sets D,~(x) form a convenient collection of neighbourhoods of points in G. Note t h a t U , + z D , ( x ) = G. C o r o l l a r y 2.25. F o r x E G,

clt 2/d= n we neglect the exponential term, a n d have

am(t) < ~ t - ~ , / ~

Z

(2-")~+~,

< ct-~lld~ (2---)2+d~ < c't21 d~. Similar calculations give the lower bound. Remarks

[]

2.26. 1. Since 2 / d ~ = log 4 / l o g 5 < 1 this implies t h a t X is subdiffusive.

2. Since ~G (B(x, r)) • r d,, for x E G, it is t e m p t i n g to t r y and prove Corollary 2.25 by the following calculation:

(2.18)

E~12~ - ~1~

=

•

/{ /o

= t 2/d~

f

r2dr

~(t,x,y)~tc(dy)

JOB(x,r)

dr r 2 rds-lt-ds/d'~ exp --c(rd~/t) l[d~'-I

(

/o

s l+a~ exp - c ( s d~)l/a~-I

(

)

)

ds = ct 2/d~.

Of course this calculation, as it stands, is not valid: the e s t i m a t e

~(B(~,~

+ ~ ) - B(x,~)) • ~ ' - ' ~

is c e r t a i n l y not valid for all r. But it does hold on average over length scales of 2 '~ < r < 2 n+l, a n d so splitting G into suitable shells, a rigorous version of this calculation m a y be o b t a i n e d - a n d this is what we did in the proof of Corollary 2.25. T h e / k - p o t e n t i a l kernel density of )~ is defined by

~,(~, y) =

e-~'~(t, ~, y) dr.

From (2.14) it follows t h a t ux is continuous, t h a t u ~ ( x , z ) 0 : )f~ = y} then

(2.19)

< oo) > 0.

It is of course clear that X must be able to hit points in Gn - otherwise it could not move, but (2.19) shows that the remaining points in G have a similar status. The continuity of u x ( x , y) in a neighbourhood of x implies that F ' ( T : = 0) = 1, that is t h a t x is regular for {x} for all x 6 G. The following estimate on the distribution of I)ft - x[ can be obtained easily from (2.14) by integration, but since this bound is actually one of the ingredients in the proof, such an argument would be circular. Proposition

2.27. For x 6 G, A > 0, t > 0,

X[ >

),)

_< c3 e x p From this, it follows that the paths of _~ are HSlder continuous of order l/d,,, for each e > 0. In fact we can (up to constants) obtain the precise modulus of continuity of X. Set h(t) = t l/d~ ( l o g t - 1 ) (d~-l)/d~ . T h e o r e m 2.28. (a) For x E G

cl _< lim

sup

8J.0 0<s O,I?a,a E Go) be a Markov chain on Go: clearly V is specified by the t r a n s i t i o n probabilities

p(al,aj) = Fa'(V1 = hi),

0 _< i,j 0, E~,x E Go) on Go which moves diagonally with probability p, and horizontally or vertically with probability 89( l - p ) . Let (Y~',r _> 0, E~,x 9 G1) be the Markov chain on G1 obtained by replication, and let T~, k > 0 be successive disjoint hits by Y' on Go.

26 0 ! T h e n writing f ( p ) = ~p(Y~'x = (1,1)) we have (after several minutes calcula-

tion) f(P)=

1 4-3p"

T h e equation f(p) = p therefore has two solutions: p = ~ and p = 1, each of which corresponds to a d e c i m a t i o n i n w r i a n t walk on Go. (The number ~ here has no general significance: if we h a d looked at the fractal similar to the Vicsek set, b u t b a s e d on a 5 x 5 square r a t h e r t h a n a 3 x 3 square, t h e n we would have o b t a i n e d a different number). One m a y now c a r r y through, in each of these cases, the construction of a diffusion on the Vicsek set F , very much as for the Sierpinski gasket. For p = 1 one gets a r a t h e r uninteresting process, which, if s t a r t e d from (0, 0), is (up to a constant t i m e change) 1-dimensionM Brownian motion on the diagonal {(t, t), 0 < t < 1}. It is w o r t h r e m a r k i n g t h a t this process is not strong Markov: for each x E F one can take I?* to be t h e law of a Brownian motion moving on a diagonal line including x, b u t the strong Markov p r o p e r t y will fail at points where two diagonals intersect, such as the point (89 89 For p = ] one obtains a process (Xt,t > 0) with much the same behaviour as t h e Brownian m o t i o n on the SG. We have for the Vicsek set (with p = 89 df(Fvs) = log 5 / l o g 3, dw(Fvs) = log 1 5 / l o g 3. This process was studied in some d e t a i l b y K r e b s [Krl, Kr2]. T h e Vicsek set was mentioned in [Go], and is one of the "nested fractals" of L i n d s t r 0 m [L1]. This e x a m p l e shows t h a t one m a y have to work to find a decimation invariant r a n d o m walk, a n d also t h a t this m a y not be unique. For the VS, one of the decimation invariant r a n d o m walks was degenerate, in the sense t h a t P~(Y hits y) = 0 for some z, y E Go, a n d we found t h e associated diffusion to be of little interest. But it raises t h e possibility t h a t there could exist regular fractals carrying more t h a n one " n a t u r a l " diffusion. T h e second example is the Sierpinski carpet (SC). For this set a more serious difficulty arises. T h e VS was finitely ramified, so t h a t if Yt is a diffusion on Fvs, a n d (Tk, k > 0) are successive disjoint hits on G,~, for some n > 0, then (YTk, k >_O) is a Markov chain on Gn. However the SC is not finitely ramified: if (Zt, t > 0) is a diffusion on Fsc, t h e n the first exit of Z from [0, 89 could occur anywhere on t h e line segments {($,y),01 _< y _< ~},1 {(z, 5),10 < x < ~}. It is not even clear t h a t a diffusion on Fsc will hit points in Gn. Thus to construct a diffusion on Fsc one will need very different m e t h o d s from those outlined above. It is possible, a n d has been done: see [BBI-BB6], a n d [Bas] for a survey. On the t h i r d question mentioned above, disappointingly little has been done: most known results on the processes on t h e Sierpinski gasket, or o t h e r fractals, are of roughly the same d e p t h as the bounds in T h e o r e m 2.23. Note however the results on t h e s p e c t r u m of/~ in [FS1, FS2, S h l - S h 4 ] , and the large deviation results in [Kumh]. Also, K u s u o k a [Kus2] has very interesting results on the behaviour of h a r m o n i c functions, which i m p l y t h a t the measure defined formally on G by

v(dx) --- I v / I z (~)~(dz) is singular with respect to p. There are m a n y open problems here.

27 3. F r a c t i o n a l D i f f u s i o n s . In this section I will introduce a class of processes, defined on metric spaces, which will include m a n y of the processes on fractals mentioned in these lectures. I have chosen an axiomatic approach, as it seems easier, and enables us to neglect (for t h e t i m e being!) much of fine detail in the geometry of the space. A m e t r i c space (F, p) has t h e midpoint property if for each x, y E F there exists 1 X , y). Recall t h a t the geodesic metric dG in z E F such t h a t p(z, z) = p ( z , y ) = ~p( Section 2 h a d this property. The following result is a straightforward exercise: L e m m a 3.1. (See [Blu]). Let (F, p) be a complete metric space with the midpoint property. Then for each x, y E F there ex/sts a geodesic path (7(t), 0 < t < 1) such that 7(0) = z, 3'(1) = y a n d p(~/(s),'r(t)) = ]t - s[d(z,y), 0 < s < t < 1. For this reason we will frequently refer to a metric p with the m i d p o i n t p r o p e r t y as a geodesic metric. See [Stul] for a d d i t i o n a l remarks a n d references on spaces of this type. D e f i n i t i o n 3.2. Let (F, p) be a complete metric space, and # be a Borel measure on ( F , B ( F ) ) . We call ( F , p , p ) a fractional metric space (FMS for short) if (3.1a)

(F, p) has the m i d p o i n t p r o p e r t y ,

a n d there exist d f > 0, and constants Cl,C2 such t h a t if r0 = s u p { p ( z , y ) : x , y E F } E (0, oo] is t h e d i a m e t e r of F t h e n (3.1b)

clr d' < # ( B ( x , r ) ) 0. A p p l y i n g the m i d p o i n t p r o p e r t y r e p e a t e d l y we obtain, for m = 2 k, k > 1, a sequence x -- zo,zl,... ,zm -- y with p(zi,zi+l) = D i m . Set r = D/2m: the balls B(zi,r) must be disjoint, or, using the triangle inequality, we would have p(x, y) < D. But then m--1

U B(z,, r) C B(x,D), i=0

so t h a t

rn--1

c2D d' > # ( B ( x , D ) ) > E

#(B(z,,r))

i=0

> mclD d~( 2 m ) - d s = cml--d~. If d f < 1 a c o n t r a d i c t i o n arises on letting m --* ~ .

[]

3.5. Let (F,p,#) be a fractional metric space. X = (~x, x E F, Xt, t ~ 0) is a fractional diffusion on F if Definition

A Markov process

(3.2a) X is a conservative Feller diffusion with state space F . (3.2b) X is # - s y m m e t r i c . (3.2c) X has a s y m m e t r i c t r a n s i t i o n density p(t, x, y) = p(t, y, x), t > 0, x, y E F , which satisfies, the C h a p m a n - K o l m o g o r o v equations a n d is, for each t > 0, j o i n t l y continuous. (3.2d) T h e r e exist constants (~,fl, 7, cl - c4, to = v0~, such t h a t

(3.3)

Clt

exp


0

lP~(p(x,X,) > T) _< Cl e x p ( - c 2 r B ' r t - ' r ) . (b) There exists c3 > 0 such that c4exp (--chT#Tt -7) < ~=(p(x, Xt) > r) forr < C3T0, t < r ~.

(c) F o r x 9 F , 0 < T < car9, /YT(X,T) = inf{s > 0 : X , • B ( z , r ) } then

(3.7)

COT~~ E~(x,r) _< C~T~.

Proof. F i x z E F , a n d set D(a, b) = {y E F : a < p(x, y) p(D(a,b)) > c3.1.1bd~ - c3.1.2adl. Choose 8 > 2 so t h a t C3.1.1od! > 2C3.1.2: t h e n we have

(3.8)

csa ~' < ~(D(a,0a))< cga~,.

Therefore, writing Dn = D(Snr, 8 n+1 r), we have # ( D n ) x 8 nd~ provided r8 n+1 _~ r0. Now

31 /.

P~(p(x,X,) > r) =

(3.9)

p(t,x,y)it(dy)

/

B(z,r) ~ = ~n~: 0 so that c3O < 1. Then It(D0) _> cr ~, and taking only the first term in (3.9) we deduce that, since r ~ > t,

~(p(~, x,) > ~) > c(~lt)~,/" exp(-c~3(~/t)~) _> c e~p(-c~ (~'/t)~). (c) Note first that

(3.10)

~(~(~,~) > t) < ~ ( x ,

e B(x,~))

= _J~(~,r)p(t, y, z)it(dz) So, for a suitable c14

_~,

yEF.

Applying the Markov property of X we have for each k _> 1

P~(T(x,r) > kel4r ~) 0,

that V > ~ (3.15)

P(~{ < t[a(~l,...,~{_l)) < P + at,

t>O.

Then (3.16)

l o g P ( V < t) O, clr ~ ~_ EX T(x,r) ~ c2r f~. Then for x C F, t > 0 ,

~(~(x,r)

< t) < (1 - el/(2'c~)) + c3r-~t.

Proof. Let x e F , a n d A = B ( x , r ) , T ----T(x,r). Since ~- _< t + (~" - t)l(~>t) we have EZ'r t ) E Y` (T -- t)

~_ t ~-Pz(T > t) supEYT. Y

As ~- ~ ~-(y, 2r) IFU-a.s. for any y C F , we d e d u c e

clr~ < E ~ < t + ~ ( ~ > t)c2(2r)~, so t h a t C22~(T

_ 0 and there exist a < b such that

/ f(x)E~ g(Yt)it(dx) = 0 for t e (a,b),

(3.18)

then f f(x)E~ g(Yt)#(dx) = 0 for all t > O.

Proof. Let (Ex, s > 0) be t h e s p e c t r a l family associated with Tt. Thus (see [FOT, p. 17]) Tt = e-XtdEx, and

I~

(f, T,g) =

/o

e-:"d(f, E ,g) =

/o

where ~, is of finite variation. (3.18) and the uniqueness of the Laplace transform i m p l y t h a t t, = O, a n d so (f, Ttg) = 0 for all t. [] 3.18. Let F a n d Y satisfy the hypotheses of Theorem 3.11. If p(x,y) < c3r 0 then ~z(Yt E B(y,r)) > 0 for all r > 0 and t > O.

Lemma

R e m a r k . T h e restriction p(z,y) < c3ro is of course unnecessary, b u t it is all we need now. T h e conclusion of T h e o r e m 3.11 implies t h a t ~ ( Y t E B(y, r)) > 0 for all r > 0 a n d t > 0, for M i x , y E F .

Proof. Suppose t h e conclusion of the L e m m a fails for x, y, r, t. Choose g E C(F, N+) such t h a t fmgdit = 1 a n d g = 0 outside B(y,r). Let tl = t/2, rl = c3(tl) ~, a n d choose f 9 C ( F , I ~ + ) so t h a t f F f d # = 1, f ( x ) > 0 and f = 0 outside A = B(x, rl). If 0 < s < t t h e n t h e construction of g implies t h a t 0

E~ g(Yt) = f~F q(s, x, x' )E x' g(Yt-,)it(dx ' ).

Since b y (3.13) q(s,x,x') > 0 for t/2 < s < t, x' 9 B(x, rl), we deduce t h a t E*'g(Yu) = 0 for x' 9 S ( x , rl), u 9 (0, t/2). Thus as s u p p ( f ) C B(x, rl)

F f (X')EX'g(Yu)dit

= 0

for all u 9 ( 1 , t / 2 ) , a n d hence, by L e m m a 3.17, for all u > 0. But by (3.13) if u = (p(x,y)/c3) ~ t h e n q(u,x,y) > 0, and by the continuity of f , g and q it follows t h a t f fE*g(Y=)eit > 0, a contradiction. []

Proof of Theorem 3.11. For simplicity we give full details of the p r o o f only in the case r o = co; the a r g u m e n t in the case of b o u n d e d 17 is essentially the same. We begin by o b t a i n i n g a b o u n d on


1, b = r/n, a n d define stopping times Si, i > 0, by So = 0,

Si+~ = inf{t _> Si : p(Ys,,Yt) _> b}.

35 Let ~i = Si - Si-1, i > 1. Let (~'t) be the filtration of Yt, and let ~i = J=s,. We have by L e m m a 3.16

~(~i+1 0 for all t. So we have fl > 1. (If r0 = 1 then we take r small enough so that r < c3). If we neglect for the moment the fact that n 9 bt, and take n = no in (3.19) so that

then

n~o-1 = (c2s/4c~)r~t -1,

(3.20) and

_< t) _< -}csn0. So if r~t -1 >_ 1, we can choose n 6 l~ so that 1 < n < no V 1, and we obtain (3.21)

I?~(T(x,r) < t) _< c9 exp

-clo

9

Adjusting the constant c9 if necessary, this bound also clearly holds if r~t -1 < 1. Now let z , y 6 F , write r = p(x,y), choose e < r/4, and set C~ = B(z,e), z = x,y. Set Ax = {z 6 F : p(z,x) < p(z,y)}, Ay = { z : p(z,x) >_ p(z,y)}. Let u~, uy be the restriction of # to C~, Cy respectively. We now derive the upper bound on q(t, x, y) by combining the bounds (3.12) and (3.21): the idea is to split the journey of Y from C~ to C~ into two pieces, and use one of the bounds on each piece. We have (3.22)

F~'(Yt 6 C,) = [ [ q(t,x',y')#(dx')#(dy') C~C.

c3t l l j we can also ensure t h a t for some cl3 > 0 (3.29)

r >_ c13(tln)l//3 '

u

so t h a t u ~-1 < 2~c~3~r~lt. So, by (3.28)

q(t,~,y) > c(tln)-~'i~c74 _> cl 5t -ds IZ exp ( n log cl4 )

>_ eist -d' i' exp

lt )'ir

) .

[]

Remarks 3.19. 1. Note t h a t the only point at which we used the "midpoint" p r o p e r t y of p is in the derivation of the lower b o u n d for q. 2. T h e essential i d e a of t h e proof of T h e o r e m 3.11 is t h a t we can obtain b o u n d s on t h e long range b e h a v i o u r of Y provided we have good enough information a b o u t the b e h a v i o u r of Y over distances of order t ll2. Note t h a t in each case, if r = p(x, y), the e s t i m a t e of q ( t , x , y ) involves splitting the j o u r n e y from x to y into n steps, where n x (r2lt) 1/(~-l). 3. B o t h the arguments for the u p p e r and lower bounds a p p e a r quite crude: the fact t h a t t h e y yield the same b o u n d s (except for constants) indicates t h a t less is thrown away t h a n might a p p e a r at first sight. T h e explanation, very loosely, is given by "large deviations". T h e off-diagonal bounds are relevant only when r 13 >> t - otherwise t h e t e r m in t h e exponential is of order 1. If r E >> t then it is difficult for Y to move from x to y by time t and it is likely to do so along more or less the shortest p a t h . T h e proof of the lower b o u n d suggests t h a t the process moves in a 'sausage' of radius t i n • t i t ~1-1. T h e following two theorems give additional bounds and restrictions on the p a r a m e t e r s d f a n d / 3 . Unlike the proofs above the results use the s y m m e t r y of the process very strongly. T h e proofs should a p p e a r in a forthcoming paper. Theorem (3.30)

3.20. Let F be a F M S ( d l ) , and X be a F D ( d f , / 3 ) on F. Then 2 O, p > O.

Since by Theorem 3.20 d", > 2 this shows that FDs are diffusive or subdiffusive. L e m m a 3.25. (Modulus of continuity). Let ~(t) = t 1/d~ (log(1/t)) (d~-D/d~. Then (3.31)

Cl < lira sup p(X~,X~) to) _< ~ ( R ~ < to) + P~(~- > to) < (1 - e -xt~ + ctodl/d~r dl. Choose first to so that the second term is less than 88 and then A so that the first term is also less than 88 We have to ~ r d~ ~ A-1, and the upper bound now follows from (3.40). The lower bound is proved in the same way, using the bounds on the lower tail of T given in (3.11). [] L e m . m a 3.35. There exist constants cl > 1, c2 such that ff x, y E F, r = p ( x , y ) , t 0 = r d~ then ~ ( T ~ < to < ~(x,c~r))> c~.

Proof. Set ~ -- (e/r) d~ ; we have px(x, y) > c3 exp(-c4e) by (3.33). So since p x ( z , y ) = E~ e -xT, c3 exp(-c4O) - exp(--0d~). As dw > 1 we can choose 0 (depending only on c3, c~ and d,o) such that P~(Tv < t) >_ ~31c exp(--c4O) ----C5. By (3.11) for a > 0

P~('r(z, an) < ~'~) c3 3~ ~ ( ~ ) . Now set r = E =' ~(X~) for x' E B. T h e n r is harmonic in B and ~ < r on B. A p p l y i n g Corollary 3.38 to r in B we deduce

r

__~r

__~C3.38A~P(Z) __~(C3.38.1)2~0(X).

Since r

-- E y (uA(y, y) -- uA(Xr, y)) the conclusion follows from (3.41).

Theorem

3.40. (a) Let ~ > 0. Then t'or x, x', y E F , a n d f E L I ( F ) , g E L ~ ( F ) ,

(3.42) (3.43) (3.44)

I ~ ( ~ , y ) - uA(x',y)l ~ clp(x,x') d~-d', IV~f(~) - v~f(~')l < ~,p(z,~')~o-d'llfll,. IV~g(~) - v~g(~')l _< c~-d./~p(~, ~,)d~-d, Ilgll~.

[]

44

Proof. Let x, x' C F, write r = p(x,x') and let R > r, A = B(x,R). uA(y, x') > pA(y, x)uA(x, x'), we have using the s y m m e t r y of X that (3.45)

=2(~, y)

-

Since

u2(~', y) < ~2(y, ~) p~(y, x)~2(~, ~') = p~(y, ~)(u~(~, x) u~(~, ~')). -

-

Thus

lu~(~,y) - ~(z',y)l -< 1~2(~,~) - u2(~,~')l. Setting ~ = O and using L e m m a 3.39 we deduce (3.46)

luA(x,y) _ ~.4(x,,y)l _< ~:d~-,~,.

So f

Iu~/(~) - UAf(~')E 0 we apply the resolvent equation in the form

u~(x, y) -- uA(x, y) - ~v%(~), where

v(x) = uA(x,y). (Note that Ilvlll = ;~-1). Thus < c : d ~ - d , + ~clrd~-~'tlvll,

2c3rd~--d:. Letting /~ --* ~ we deduce (3.42), and (3.43) then follows, exactly as above, by integration. To prove (3.46) note first that p~(y,x) = ux(y,x)/ux(x,x). So by (3.33) (3.47)

f p2(y,~)lf(y)l.(dy)

0). Let ~o(u) = u(d~-d')/2(log(I/u) ) I/2. The modulus of continuity in space of L" is given by: l i m sup ~Oo<s 0, we have IP~~ < 1) > 0. So oo > EX~

> EZ~

a n d thus MTD < ar a.S. So u~(XTD,Yi) < Or for each Yl E D, and thus we must have u~(yi, Yi) < ~ for some y~ E D. So, by Proposition 3.25, ds < 2. [] R e m a r k 3.43. For k = 1,2 let (Fk,dk,#k) be F M S with dimension d r ( k ) , a n d c o m m o n d i a m e t e r r0. Let F = F1 • F2, let p _> 1 and set d((xl,x2),(yl,Y2)) = (dl(xl,Yl) p + d2(x2,Y2)P) Up, # = #1 • #2. T h e n (F,d,#) is a F M S with dimension d f = d f ( 1 ) + d f ( 2 ) . Suppose t h a t for k = 1,2 X k is a FD(dl(k),dw(k)) on Fk. T h e n if X = ( X ~ , X 2) it is clear from t h e definition of F D s t h a t if dw(1) = d,~(2) = / 3 t h e n X is a F D ( d f , f l ) on F . However, if dw(1) r dw(2) then X is not a F D on F . (Note from (3.3) t h a t the metric p can, up to constants, be e x t r a c t e d from the t r a n s i t i o n density p(t, x, y) by looking at limits as t ~ 0). So the class of F D s is not stable u n d e r products. This suggests t h a t it might be desirable to consider a wider class of diffusions with densities of the form: Tt

1

where Pi are a p p r o p r i a t e non-negative functions on F • F . Such processes would have different space-time scalings in the different 'directions' in the set F given by t h e functions pi. A recent p a p e r of H a m b l y and K u m a g a i [HK2] suggests t h a t

46 diffusions on p.c.f.s.s, sets (the most general type of regular fractal which has been studied in detail) have a behaviour a little like this, though it is not likely that the transition density is precisely of the form (3.48).

Spectral properties. Let X be a FD on a FMS F with diameter r0 = 1. The bounds on the density

p(t, x, y) imply that p(t,., .) has an eigenvalue expansion (see [DaSi, Lemma 2.1]). T h e o r e m 3.44. There exist continuous functions ~ol, and Ai with 0 0 define Ea on 7) by E a ( f , f ) -~ e ( f , f ) +

llfll~ = llfll~+ c,E(f,f)

=

write

E,~(I,f).

D e f i n i t i o n 4.2. Let (s 7)) be a symmetric form. (a) E is closed if (79, II-Ile~) is complete (b) (s is Markov if for f E 7), i f g = ( 0 V f ) A 1 t h e n g E 79 and s < E(f,f). (c) (s 7)) is a Dirichlet form if 7) is dense in LZ(F, #) a n d (s 7)) is a closed, Markov s y m m e t r i c form. Some further p r o p e r t i e s of a Dirichlet form will be of importance: D e f i n i t i o n 4.3. (E, 7)) is regular if (4.5) (4.6)

79 N Co(F) is dense in 7) in II" I1~1, a n d 7) N Co(F) is dense in Co(F) in [[. I1~.

s is local if C(f, g) = 0 whenever f, g have disjoint support. E is conservative if 1 E 7) and s = 0. s is irreducible if s is conservative and g ( f , f ) = 0 implies t h a t f is constant. T h e classical example of a Dirichlet form is t h a t of Brownian motion on Ra:

s

f) - -

1 ~

f IVfl2 d~, I e H1'2(R% ~a

L a t e r in this section we will look at the Dirichlet forms associated with finite state Markov chains. J u s t as the Hille-Yoshida theorem gives a 1 - 1 correspondence between semigroups a n d their generators, so we have a 1 - 1 correspondence between Dirichlet forms a n d semigroups. Given a semigroup (Tt) the associated Dirichlet form is o b t a i n e d in a fairly straightforward fashion. D e f i n i t i o n 4.4. (a) T h e semigroup (Tt) is iarkovian if f E L2(F,#), 0 0) be a strongly continuous It-symmetric contraction semigroup on L2(F, It), which is Markovian. For f C L 2 ( F , # ) the function ~l(t) defined by Theorem

q o f ( t ) = t - x ( f --Ttf, f),

t>0

is non-negative and non-increasing. Let V = {f 9 L 2 ( F , # ) : lim~oy(t) < oo}, tlo

C(Lf) =limbos(t), tlo

f e 7).

Then (s 7)) is a Dirichlet form. If (Z:, 79(/:)) is the infinitesimal generator of (Tt), then 7)(1:) C V, V(C) is dense in L2(F, It), and s

(4.7)

= ( - E l , g),

f E 7)(s

g e 7).

As one might expect, by analogy with the infinitesimal generator, passing from a Dirichlet form (s 7)) to the associated semigroup is less straightforward. Since formally we have Us = (a - 12)-1, the relation (4.7) suggests that (4.8)

( f , g ) = ((a - E)U~f,g) -- a(U~,f,g) + E(Usf, g) = C~(UsI, g).

Using (4.8), given the Dirichlet form E, one can use the Riesz representation theorem to define U~f. One can verify that Us satisfies the resolvent equation, and is strongly continuous, and hence by the Hille-Yoshida theorem (Us) is the resolvent of a semigroup (Tt). T h e o r e m 4.6. ([FOT, p.18]) Let (e,7)) be a Oirichlet form on L2(F,#). Then there exists a strongly continuous #-symmetric Maxkovian contraction semigroup (Tt) on L2(F,#), with infinitesimal generator (L:,7)(E)) and resolvent (U~,a > O)

such that 1: and ~ satisfy (4.7) and also (4.9)

s

+ a(f,g) -- (f,g),

f e L2(F, It), g e 7).

Of course the operations in Theorem 4.5 and Theorem 4.6 are inverses of each other. Using, for a moment, the ugly but clear notation C -- T h m 4.5((Tt)) to denote the Dirichlet form given by Theorem 4.5, we have T h m 4.6(Tam 4.5((Tt))) = (Tt), and similarly T h m 4.5(Whm 4.6 (s

= E.

R e m a r k 4.7. The relation (4.7) provides a useful computational tool to identify the process corresponding to a given Dirichlet form - at least for those who find it more natural to think of generators of processes than their Dirichlet forms. For example, given the Dirichlet form s f) = f IV f[ 2, we have, by the Gauss-Green formula, for f , g e C2(~d), ( - s =s = f v f.Vg = -- f g A f , so that s = A. We see therefore that a Dirichlet form (~,7)) give us a semigroup (Tt) on L2(F, It). But does this semigroup correspond to a 'nice' Markov process? In general it need not, but if C is regular then one obtains a Hunt process. (Recall that

50 a Hunt process X = (X,, t > 0, ~ , x E F ) is a strong Markov process with cadlag sample paths, which is quasi-left-continuous.) T h e o r e m 4.8. ([FOT, Thin. 7.2.1.]) (a) Let (~, 9 ) be a regu/ar Dirichlet form on L 2 ( F , # ) . Then there exists a #-symmetric Hunt process X = ( X t , t > O , ~ , x E F ) on F with Dirichlet form C. (b) In addition, X is a diffusion if and only i r e is local. R e m a r k 4.9. Let X = (X~,t >_ 0,I?~,x E I~2) be Brownian motion on R 2. Let A C I~2 be a polar set, so that P~(TA < oo) = 0 for each x. T h e n we can obtain a new Hunt process Y -- (X~ >_ 0, Q=, x E ]~') by "freezing" XonA. SetQ= =~*,xEA c,andforzEAletQ~(xt=x, a l l t E [0, oo)) = 1 . Then the semigroups (TX), (TY), viewed as acting on L2(~2), are identical, and so X and Y have the same Dirichlet form. This example shows that the Hunt process obtained in Theorem 4.8 will not, in general, be unique, and also makes it clear that a semigroup on L 2 is a less precise object t h a n a Markov process. However, the kind of difficulty indicated by this example is the only problem - - see [FOT, Thm. 4.2.7.]. In addition, if, as will be the case for the processes considered in these notes, all points are non-polar, then the Hunt process is uniquely specified by the Dirichlet form g. We now interpret the conditions that ~ is conservative or irreducible in terms of the process X. L e m m a 4.10. I_fs is conservative then Ttl = 1 and the associated Markov process X has infinite lifetime.

Proof. If f e 9 ( s then 0 cllfl[~ +~/~, f e v.

This inequality appears awkward at first sight, and also hard to verify. However, in classical situations, such as when the Dirichlet form E is the one connected with the Laplacian on ]~d or a manifold, it can often be obtained from an isoperimetric inequality.

51 In w h a t follows we fix a regular conservative Dirichlet form (E, V). Let (Tt) be the associated semigroup on L 2 (F, #), and X = (Xt, t > 0, P~) be the Hunt process associated with E. 4.12. ([CKS, Theorem 2.1]) (a) Suppose E satisfies a Nash inequality ~ t h constants c, 6, 8. Then there exists c' = c'(c, 8) such that

Theorem

(4.11)

[[Ttlh-~ < c ' d q - ~

t > O.

(b) If (Tt) satisfies (4.11) with constants c', 6, 0 then C satisfies a Nash inequality with constants c" = c"(c', 8), 6, and O. Proof. I sketch here only (a). Let f Z :D(s

T h e n writing ft = Ttf, and

gth : h-l(/t+h - ft) - T t s

we have llg~hll~ < IIg0hll2 ~ 0 as h ~ 0. It follows that (d/dt)ft exists in L 2 ( F , ~ ) and t h a t

d

-~ft = T t s

= s

Set ~(t) = (ft, ft). T h e n

h -1 (T(t + h) - T(t)) - 2(TtEf, T t f )

= (gth, ft + ]t+h) + (TtEf, ~t+h -- ft),

a n d therefore T is differentiable, a n d for t > 0 (4.12)

~o'(t) = 2(/25, ft) = - 2 E ( f t , ft).

If f C L2(F,t~), T t f E :D(s for each t > 0. So (4.12) extends from f E :P(s

to all

f e L2(F,#). Now let f > 0, a n d Hf]]l = 1: we have ]1s

(4.13)

~'(t) -- -2c(1,, ft)) < 2~11/~11~ cllI~ll~ +4/~ = 2~i~o(t)2 -

Thus ~ satisfies a differential inequality. Set r r If r

= 1. T h e n by (4.10), for t > 0,

is the solution of r

< -2cr176

-

c~~ 1+2/~

= e-2~t~o(t). T h e n

e46t/~ < - 2 c r

1+2/~

.1+~./0 = -c~v 0 then for some a E ]~ we have, for co = co(c, 0),

Co(t) =- co(t -4- a) -0/2. If r

is defined on (0, oc), t h e n a > 0, so t h a t < cot -~

r

t > o.

It is easy to verify t h a t r satisfies the same b o u n d - so we deduce t h a t

(4.14)

IlTdll~ ~-- e2~t~b(t) O, x , y E F' • F ~, such that P t ( x , A ) = f p ( t , x , y ) y ( d y ) for x E F',

p(t,x,u)

t > O, A E B ( F ) ,

A = p ( t , u , x ) rot all x , u , t ,

p(t + s,~,z) = [ p ( s , z , y ) v ( t , y , z ) , ( d y ) J

for all

x,z,t,s.

53 C o r o l l a r y 4.15. Suppose (E,/3) satisfies a Nash inequality with constants c, 6, O. Then, for all x , y E F ~, t > O,

p(t, x , y ) __Ty : Xt = x} and S is d e l e d

~ =

f

similarly, we have

1G(xs) ds.

However by Doeblin's theorem for the stationary measure of a Markov Chain

(4.29/

.(a) = (E~S/-1E 9

f 1G(X.lds~(F).

Rearranging, we deduce that

= (#(F)/#(G))E~S

= (.(F)/~(G))(E~ + E~.) = R.(F).

[]

C o r o l l a r y 4.28. Let H C F, z ~ H. Then

E ' T H 1,

F= U wEW~

If a = ( a l , . . . , a M ) is a vector indexed by I, we write (5.9)

a~ = l ~ a~,,,

w E Wn.

i~l

Write A~ = r m > n) write (5.10)

for w E UnW,~, A C F. If n_> 1, a n d w E W (or W,n with w i n -= ( w l , . . . ,wn) E Wn.

L e m m a 5.9. For each w E W, there ex/sts a x ~ E F such that (5.11)

fi r

= {x~}.

n----1

Since r = Cw]~(r C r the sequence of sets (5.11) is decreasing. As r are continuous, r are compact, and therefore A N~F~I n is non-empty. But as diam(F~ln) _< (1 - 5)~diam(f), we have diam(A) = so that A consists of a single point. Proof.

in = 0, []

63 Lemma

5.10. There exists a unique map Ir : W ~ F such that

(5.12)

~(i.w) : r

w ~ W,

i ~ I.

lr is continuous and surjective. Proof. Define ~r(w) : zw, where z~, is defined by (5.11). Let w E W. T h e n for any n,

r ( i - w) E F(i.w)l, = Fi.(~l,~_i ) --- r

).

So 7r(i. w) E Amr = {r proving (5.12). If r' also satisfies (5.12) t h e n ~d(v. w) = r (Tr'(w)) for v E Wn, w E W, n > 1. T h e n r ' ( w ) E F~,in for any n > 1, SO

71" ! ~

"ft.

To prove t h a t 7r is surjective, let x E F . By (5.7) there exists Wl E IM such that x E F= 1 = r M = U~o~=zF~I~ 2. So there exists w2 such t h a t x E F ~ 2, and continuing in this way we o b t a i n a sequence w -- ( w l , w 2 , . . . ) E W such t h a t z E F,~ln for each n. It follows t h a t x = ~r(w). Let U be open in F , a n d w E 7r-l(U). T h e n F=ln M U c is a decreasing sequence of c o m p a c t sets w i t h e m p t y intersection, so there exists m with F~,[m C U. Hence V -- {v E W : v[m --- w[m} C 71"-l(U), and since V is open in W, r - l ( U ) is open. Thus lr is continuous. [] Remark (5.13) Lemma

5.11. It is easy to see t h a t (5.12) implies t h a t r(v-w) = r

v E W,,

w ~ W.

5.12. For x E F, n >_ O set

N,(x) = U{F= : w E W,,z

E F~,}.

Then { N , ( x ), n > 1} form a base of neighbourhoods of x. Proof. Fix x and n. If v E W~ and x r F+ then, since F . is compact, d(x,F~) = i n f { d ( x , y ) : y E F~} > 0. So, as Wn is finite, d(x, N n ( x ) c) = min{d(x,F~) : x r F~,v E W~} > 0. So x E i n t ( N n ( x ) ) . Since d i a m F w < ( 1 - 6 ) n d i a m ( F ) f o r w E W,~ we have d i a m N , ( x ) < 2(1 - 6 ) ' ~ d i a m ( f ) . So if g ~ x is open, N~(x) C U for all sufficiently large n. [] T h e definition of a self-similar structure does not contain any condition to prevent overlaps between the sets r i E IM. (One could even have r -- r for example). For sets in I~d t h e open set condition prevents overlaps, b u t relies on t h e existence of a space in which the fractal F is embedded. A general, a b s t r a c t , non-overlap condition, in terms of dimension, is given in [KZl]. However, for finitely ramified sets the s i t u a t i o n is somewhat simpler. For a self-similar s t r u c t u r e S = (F, r i E IM) set

B = B(S) =

U i,j,i~ j

n Fj.

64 As one might expect, we will require B ( S ) to be finite. However, this on its own is not sufficient: we will require a stronger condition, in terms of the word space W. Set Y = rc-1 (B(S)), P = 0

~r~(F)"

n=l

D e f i n i t i o n 5.13. A self-similar structure (F, r is post critically finite, or p.c.f., if P is finite. A metric space ( F , d ) is a p.c.f.s.s, set if there exists a p.c.f, self-similar structure (r 1 < i < M ) on F. R e m a r k s 5.14. 1. As this definition is a little impenetrable, we will give several examples below. The definition is due to Kigami [Ki2], who called F the critical set of 8, and P the post critical set. 2. The definition of a self-similar structure given here is slightly less general than that given in [Ki2]. Kigami did not impose the constraint (5.6) on the maps r but made the existence and continuity of 7r an axiom. 3. The initial metric d on F does not play a major role. On the whole, we will work with the natural structure of neighbourhoods of points provided by the self-similar structure and the sets F,~, w E W ~ , n >_ O. E x a m p l e s 5.15. 1. T h e Sierpinski gasket. Let al, a2, a3 be the corners of the unit triangle in IRd, and let r

= ai + 89 - a,),

x e R 2,

1l.

5 . 2 0 . L e t s 9 { 1 , . . . , M } . T h e n r(~) is in e x a c t l y one n-complex, for each

Proof. Let n -- 1, a n d write x, = 7r(~). P l a i n l y x, 9 F , ; suppose x, E Fi where i # s. T h e n x~ -- r for some w 9 W. S i n c e x , -- r for a n y k_> 1, x~ = r (1r(i.w)) = 7r(s k . i . w ) , where s k = (s, s . . . . . s) 9 Wk. Since x , 9 F, MF, C B , z c - l ( x , ) 9 C. B u t therefore s k 9 i . w 9 C for each k > 1, a n d since i # s, C is infinite, a c o n t r a d i c t i o n . Now let n > 2, a n d s u p p o s e x , = ~r(~) 9 F ~ , where w 9 W,~ a n d w # s n. Let 0 < k < n - 1 b e such t h a t w = s k . a k w , a n d Wk+l # s. T h e n a p p l y i n g r to F,~ we have t h a t x, 9 F~,~w M F , ~ - ~ , which c o n t r a d i c t s the case n = 1 above. [] Let ( F , r

,r

be a p.c.f.s.s, set. For x 9 F , let

ran(x) = # {w e Wn : x e Fw} be t h e n - m u l t i p l i c i t y of x, t h a t is t h e n u m b e r of d i s t i n c t n - c o m p l e x e s c o n t a i n i n g x. P l a i n l y , i f x r U,~V (n), t h e n m n ( x ) = 1 for all n. Note also t h a t m . ( x ) is increasing.

Proposition

5 . 2 1 . For a11 x E F , n >_ 1, m,(x) < M#(P).

Proof. S u p p o s e x E Fwl M . . . M Fwh, where w i, 1 < i < k are d i s t i n c t e l e m e n t s of W,). S u p p o s e first t h a t w~ # w~ for some i # j . T h e n x E B, a n d therefore there exist v l , . . . , v k C W such t h a t ~r(wl . v t) -- x, 1 < l < k. Hence wt . v I E r for each l, a n d so # ( r ) _> k. B u t # ( P ) > M - I # ( F ) , a n d thus k 1

wGW~

Proof. It is sufficient to prove (5.20) in the case n : 1: the general case then follows by iteration. Write G -- F - V (~ Note that G~ M G~ = ~ if v, w E Wn a n d v ~ w. As # is non-atomic we have # ( F ~ ) = # ( G ~ ) for any w E Wn. Let f = l a ~ for some w E W~. T h e n f o r = 0 if i ~ Wl, and f o r = 1 G ~ . Thus

fIfo

0=:f fd ,

proving (5.20) for this particular f . The equality then extends to L 1 by a s t a n d a r d argument. [] We will also need related measures on the sets V (n). Let No -- # V (~ and set (5.21)

#~(x) = N0-1 ~ 0~lv(o,(X), wcw~

Fix

x E Y (~).

L e m m a 5.29. /z,~ is a probability measure on V ('~) a n d w l i m , ~ o o # n -- #e.

Proof. Since # V (0) -- No we have

~(v(~)) : Z No' Z e=~,o~(~): Z e~:~, zEV(~)

wGW~

wEW~

proving the first assertion. We may regard #n as being derived from # by shifting the mass on each ncomplex F ~ to the b o u n d a r y V(~ with an equal amount of mass being moved to

74 each point. (So a point x E V(~ obtains a contribution of 0,~ from each n-complex it belongs to). So if f : F --~ R then (5.22)

f fd#JF

f fd#n JR

_< max sup I f ( x ) - f(Y)l wEW~ x,yEF~

It follows t h a t #n--*#0.~ Symmetries of p.c.s

[] sets.

D e f i n i t i o n 5.30. Let g be a group of continuous bijections from F to F. We call G a symmetry group of F if (1) g : V (~ ~ V (~ for all g E G. (2) For each i E I, g E G there exists j E I, g~ E G such that (5.23)

g

o

r

= e j o g'.

Note that if g, h satisfy (5.23) then (goh) or162162 =

(r

o

g') o h'

=

ek o g",

for some j , k E I, g',h*,g" E ~. The calculation above also shows that if G1 and G2 are s y m m e t r y groups then the group generated by G1 and G2 is also a s y m m e t r y group. Write G(F) for the largest symmetry group of F. If G is a symmetry group, and g E G write ~(i) for the unique element j E I such that (5.23) holds. L e m m a 5.31. Let g E G. Then for each n _> 0, w E W,~, there exist v E W~, g' E G such that g o ev~ = ev o g'. In particular g : V ('q --* V (n). Proof. The first assertion is just (5.23) if n = 1. If n _> 1, and the assertion holds for all v E W,~ then if w = i 9v E W,,+I then

gor for

j E I, gl,g. E ~.

= gor

or

= ~ og' o r

= ~j o r

og",

[]

Proposition

5.32. Let (F, r . . , ~)M) be an A N F . Let G1 be the set of isometries of R a generated by reflections in the hyperplanes bisecting the line segments [zi, zj ], i ~ j, z,, zj E V (~ Let Go be the group generated by G1. Then GR = {giF : g E GO} is a s y m m e t r y group of F. Proof. I f g E G1 t h e n g : V (n) --~ V (n) for each n and hence a l s o g : F - - ~ F. Let i E I: by the s y m m e t r y axiom (A2) g(Vi(~ = V(~ for some j E I. For each of the possible forms of V (~ given in Remark 5.25(3), the symmetry group of V (~ is generated by the reflections in G1. So, there exists g' E G0 such that g o r = r o g'. Thus (5.23) is verified for each g E G~, and it follows that (5.23) holds for all g E G0. [] R e m a r k 5.33. In [BK] the collection of 'p.e.f. morphisms' of a p.c.f.s.s, set was introduced. These are rather different from the symmetries defined here since the definition in [BK] involved 'analytic' as well as 'geometric' conditions.

75 Connectivity Properties. D e f i n i t i o n 5.34. Let F be a p.c.f.s.s, set. For n > 0, define a graph structure on V (n) by taking {x, y} E En if x # y, and x, y E V~(~ for some w E W=. P r o p o s i t i o n 5.35. Suppose that (V (1), E1 ) is connected. Then (V(n), En) is connected for each n > 2, and F is pathwise connected. Proof. Suppose that ( v ( n ) , E n ) is connected, where n > 1. Let x , g E V ('~+1). If x , g E V (1) for some w E Wn, then, since (V(1),E1) is connected, there exists a path r = z o , z l , . . . ,zk = r in (V(1),E1) connecting r and r We have zi-1, zi E v(~ ,wl for some wi E W1, for each 1 < i < k. Then if z[ = r ! zi_ 1, z iI E Fw~.~ and so {z~_l,zl } E En+l. Thus x , y are connected by a path in (V (=+~),E.+I). For general x, y E V (n+l), as (V('~),En) is connected there exists a path Yo,...,Y,~ in ( v ( n ) , E n ) such that { Y i - l , y i } E En and x,yo, and Y, Ym, lie in the same n + 1-cell. Then, by the above, the points x, Y0, Y l , . . . , Ym, Y can be connected by chains of edges in En+l. To show that F is path-connected we actually construct a continuous path 7 : [0, 1] --~ F such that F = {7(t),t C [0, 1]}. Let x 0 , . . . ,XN be a path in (V(I),E1) which is "space-filling", that is such that V (1) C { x 0 , . . . , XN}. Define 7 ( i / N ) = x~, A1 = { i / N , 0 j. T h e n z~, 0 < i < k is a p a t h in ( V ( n ) , E n ) connecting x and y', and so [] dn(x, y) = k > d , ( x , y') = an. Lemma

5.40. Let x, y E V (n) 9 Then for m > 0

(5.24)

a m d , ( x , y) < d , + m ( x , y) _ an. T h e n since zi~_l,zij lie in the same n-cell, ij - ij-1 = dm(zi~_l,zij) >_ am, by L e m m a 5.39. So r = E ; : I ( i j - i j - 1 ) -> a , am. [] C o r o l l a r y 5.41. There exists 7 E [L, boa1] such that (5.26)

bo17 n _< an _< 7".

Proof. Note t h a t log(b0an) is a subadditive sequence, and t h a t log an is superadditive. So by the general t h e o r y of these sequences there exist 00, 01 such t h a t 00 = l i m n -1 log(b0an) = inf n -1 log(b0a~), 01 = l i m n -1 l o g ( a , ) = sup n -1 l o g ( a , ) . n---* o o

n~0

So 00 = 0x, a n d setting 7 -- ee~ (5.26) follows. To o b t a i n b o u n d s on ~/ note first t h a t as a , _ O, w E Wn, (5.27)

dF(x, y) (_ el"/-n for x, y E Fw,

and (5.28)

dF(X, y) > c2~/-n for x E V (n), y E N n ( x ) c.

(b) d e induces the same topology on F as the Euclidean metric.

78 (c) dF has the midpoint property, (d) The Hausdorff dimension of F with respect to the metric dF is log M

d r ( F ) - - log 7

(5.29)

Proof. W r i t e V = UnV(n). By L e m m a 5.41 for x, y E V we have (5.30)

bo17md,~(x, y) < dn+m(x, y) < boTmdn(x, y).

So (7-"~dn+m(x,y),m _> 0) is b o u n d e d above and below. By a diagonalization a r g u m e n t we can therefore find a subsequence nk ~ cx) such t h a t

d f ( x , y ) = lim 7 - ' ~ d , ~ ( x , y ) exists for each x,y 9 V. k---~ o o

So, if x, y c V(~ where w 9 W~ then

(5.31)

Co17 - ~ < dF(x, y) < c07 - ~

It is clear t h a t dR is a metric on V. Let n > 0 and y 9 V ('~). For m = n - 1 , n - 2 , . . . , 0 choose inductively Ym 9 V (m) such t h a t Ym is in the same m-cell as Y,~+I,...,Y,. T h e n _ max{dl(X',y') : z', d,,~+l(ym,Ym+l)
0. We also choose a vector r : ( r l , . . . , rM) of positive "weights": loosely speaking ri is the size of the set r : Fi, for 1 < i < M . We call r a resistance

vector. D e f i n i t i o n 6.1. Let D be the set of Dirichlet forms C defined on C(V(~ From Section 4 we have t h a t each element C 9 D is of the form CA, where A is a conductance m a t r i x . Let also I~ 1 be the set of Dirichlet forms on C(V(1)). We consider two operations on D: (1) R e p l i c a t i o n - i.e. extension of s 9 I~ to a Dirichlet form C n 9 lI)l. (2) D e c i m a t i o n / R e s t r i c t i o n / T r a c e . Reduction of a form C 9 D1 to a form ~ 9 D. N o t e . In Section 4, we defined a Dirichlet form (C, :D) with domain 9 C L 2 (F, #). But for a finite set F , as long as p charges every point in the set it plays no role in the definition of s We therefore will find it more convenient to define E on C(F) = { f : f - ~ R}.

D e f i n i t i o n 6.2. Given s 9 D, define for f,g 9 C(V(1)),

M

CR(f,g) = ~ r~-lC(f o r

(6.2)

or

i----1 (Note t h a t as r

: V (~ --* V (1), / o r

9 C(V(~

Define R : ]D --~ D1 by

R(c) = c R

L e m m a 6.3. Let C = CA, and let M

(6.3)

R

axy : ~

--1

l(zEVi(0))l(yEVi(o,)r i ar

).

i=1

Then (6.4)

Cn(f,g) = ~12 a~,n(f(x) - f(y))(g(x) - g(y)).

80 A R = (a~y) R is a c o n d u c t a n c e m a t r i x , and s

is the associated Dirichlet form.

R _> 0 i f x ~ y , a n d a ~R < 0 . are injective, it is clear t h a t a~y R = ay~ R is i m m e d i a t e from the s y m m e t r y of A. Writing xi = r (x) we have Also azy Proof. As the m a p s r

E

ely:

~-~r/-llv(o)(X)~

yEV(1)

1v(o,(y)ar162

i

yEV(1)

i

yEv(O)

)

so A R is a conductance matrix. To verify (6.4), it is sufficient by linearity to consider the case f -- g = 6~, z 9 Y (I). Let B = {i 9 W1 : z 9 Vi(~ I f i r B, then f o r cannot equal z. If i 9 B, then f o r = 6~,(z), where zi : r s

o

r

f

o

r

sincer So,

= C(6,,, ~z,) ----- a z , z,.

Thus s

f ) = -- ~

M r ~ l a z i z , = -- ~ ri-11vi(O)(z)acrl(z),erl(z )

iEB

=

R

--azz

~

i=l

while

X:

Zc~

So (6.4) is verified.

_

S(y))

_fTAR f

R

[]

T h e most intuitive explanation of the replication operation is in terms of electrical networks. T h i n k of V (~ as an electric network. Take M copies of V (~ and rescale the i t h one by multiplying the conductance of each wire by r~-1. (This explains why we called r a resistance vector). Now assemble these to form a network with nodes V (1), using the i t h network to connect the nodes in V~(~ T h e n g R is the Dirichlet form corresponding to the network V (1). As we saw in the previous section, for x , y E VO) there m a y in general be more t h a n one 1-cell which contains b o t h x and y: this is why the sum in (6.3) is necessary. If x a n d y are connected by k wires, with conductivities c l , . 9 9 ck t h e n this is equivalent to connection by one wire of conductance c1 + .. 9 + c~. R e m a r k 6.4. T h e replication of conductivities defined here is not the same as the replication of t r a n s i t i o n probabilities discussed in Section 2. To see the difference, consider again the Sierpinski gasket. Let V (~ -- {Zl,Z2, z3}, and Y3 be the midp o i n t of IZl,Z:], a n d define y~, Y2 similarly. Let A be a conductance m a t r i x on V (~ a n d write aij = a ~ z j . Take r l : r2 = r3 = 1. While the continuous time Markov Chains X (~ X (1) associated with s and CA R will d e p e n d on the choice of a measure on V (~ and V (D, their discrete time skeletons t h a t is, the processes X (i)

81

sampled at their successive j u m p times do not - see Example 4.21. Write these processes. We have ~Ys

(]i1(

1) E {Z2, Yl

})

y(i)

for

a12 + a31 = 2a12 + a31 + a23

On the other hand, if we replicate probabilities as in Section 2,

in general these expressions are different. So, even when we confine ourselves to symmetric Markov Chains, replication of conductivities and transition probabilities give rise to different processes. Since the two replication operations are distinct, it is not surprising that the dynamical systems associated with the two operations should have different behaviours. In fact, the simple symmetric random walk on V (~ is stable fixed point if we replicate conductivities, but an unstable one if we replicate transition probabilities. The second operation on Dirichlet forms, that of restriction or trace, has already been discussed in Section 4. Definition

6.5. For E 9 D1 let

(6.5)

T(s

Define A : D --* D by A(8) = that if O > 0,

=

T(R(8)).

Tr(EIV(~ Note that A is homogeneous in the sense

A(SE) = 8A(s E x a m p l e 6.6. (The Sierpinski gasket). Let A be the conductance matrix corresponding to the simple random walk on V (~ so that a~y=l,

x~y,

a~ =-2.

Then A R is the network obtained by joining together 3 symmetric triangular networks. If A(EA) = CB, then B is the conductance matrix such that the networks (V(1),A R) and (V(~ are electrically equivalent on V (~ The simplest way to calculate B is by the A - Y transform. Replacing each of the triangles by an (upside down) Y, we see from Example 4.24 that the branches in the Y each have conductance 3. Thus (V (1), A R) is equivalent to a network consisting of a central triangle of wires of conductance 3/2, and branches of conductance 3. Applying the transform again, the central triangle is equivalent to a Y with branches of conductance 9/2. Thus the whole network is equivalent to a Y with branches of conductance 9/5, or a triangle with sides of conductance 3/5. Thus we deduce A(EA) = EB,

where

3 B = ~A.

82 T h e example above suggests t h a t to find a decimation invariant r a n d o m walk we need to find a Dirichlet form s E ID such t h a t for some A > 0 (6.6)

h(s

= As

Thus we wish to find an eigenvector for the m a p A on ]I). Since however (as we will see shortly) A is non-linear, this final formulation is not p a r t i c u l a r l y useful. Two questions i m m e d i a t e l y arise: does there always exist a non-zero (E, A) satisfying (6.6) a n d if so, is this solution (up to constant multiples) unique? We will abuse terminology slightly, a n d refer to an E E I]) such t h a t (6.6) holds as a fixed point of A. (In fact it is a fixed point of A defined on a quotient space of :D.) Example

6.7. ("abc gaskets" - see [HHW1]).

Let m l , m2, m3 be integers with mi _> 1. Let Zl, z2, za be the corners of the unit triangle in R 2, H be the closed convex hull of {zl, z2, z3 }. Let M = m l + m2 + ma, and let r 1 < i < M be similitudes such t h a t (writing for convenience eM+j = ej, 1 < j < M ) Hi = r C H , and the M triangles Hi are arranged round the edge of H , such t h a t each triangle Hi touches only H i - a a n d H i + l . (//1 touches HM and H~. only). In addition, let za E / / 1 , z2 E Hm~+l, za E H m , + m l + l . So there are ma + 1 triangles along the edge [zl,z2], and m l + 1, m2 + 1 respectively along [z~,z3], [z3,zl]. We assume t h a t r are rotation-free. Note t h a t the triangles /-/2 and HM do not touch, unless m l = m2 = m3 = 1. Let F be the fractal o b t a i n e d by Theorem 5.4 from ( r 1 6 2 To avoid unnecessarily c o m p l i c a t e d n o t a t i o n we write r for b o t h r and r

Figure 6.1: abc gasket with mx = 4, m2 = 3, m3 = 2. It is easy to check t h a t (F, r , eM) is a p.c.f.s.s, set. Write r = 1, s = m 3 + 1 , t = m3 + m , + 1. We have ~(/~) = ~ ( ( / + 1)§ for 1 < i < m3, ~(ii) = ~ ( ( i + 1)~) form3+l 0, we deduce t h a t ~ = ~(f(a)) = Aft. So, from (6.7), ~2~~2 A-I~{-I = Z;-I + ( ~ + ),.,s,

which implies t h a t ) - 1 > 1. Writing T = ] ~ 1 ~ 2 f l 3 / S , and 0 = TA(1 - A ) - ' , we therefore have ml(Z2 + 8 3 ) = 0, a n d (as (6.8)

S,T

are s y m m e t r i c in the ~i) we also o b t a i n two similar equations. Hence f12+83-0/ml,

fl3+fl, =8/m2,

fl1+82 =0/m3,

which has solution (6.9)

281 =

O(m~ 1 + m31 - m~l),

etc.

84 Since, however we need the/~i > 0, we deduce that a solution to the conductivity renormalization problem exists only if m~-1 satisfy the triangle condition, that is that (6.10)

m~- l + m ~ -1 > m l 1,

m31+ml

I >m21,

m~- l + m 2 1

> m ~ -1.

If (6.10) is satisfied, then (6.9) gives fl~ such that the associated c~ = ~ - l ( f l ) does satisfy the eigenvalue problem. In the discussion above we looked for strictly positive ~ such that ~(~) -~ ha. Now suppose that just one of the a~, ~3 say, equals 0. Then while z, and z2 are only connected via z3 in the network V (~ they are connected via an additional path in the network V (1). So, ~(a)3 > 0, and ~ cannot be a fixed point. If now al > 0, and a2 = a3 = 0 then we obtain ~(a)2 -- ~(~)3 : 0. So ~ = (1, 0, 0) satisfies ~(a) = h a for some A > 0. Similarly (0, 1, 0) and (0, 0, 1) are also fixed points. Note that in these cases the network (V (~ A(a)) is not connected. The example of the abc gaskets shows that, even if fixed points exist, they may correspond to a reducible (ie non-irreducible) E E ~). The random walks (and limiting diffusion) corresponding to such a fixed point will be restricted to part of the fractal F. We therefore wish to find a non-degenerate fixed point of (6.6), that is an s E I]) such that the network (V (~ A) is connected. D e f i n i t i o n 6.8, Let ]])~ be the set of E E D0 such that s is irreducible - that is the network (V(~ is connected. Call E E]I) strongly irreducible if C -- s and a ~ > 0 for all x :~ y. Write D ~ for the set of strongly irreducible Dirichlet forms on V(~ . The existence problem therefore takes the form: P r o b l e m 6.9. (Existence). Let ( F , r 1 6 2 Does there exist E EI]) i, A > 0, such that

(6.12

A(E)

be a p.c.f.s.s, set and let ri > 0.

:

Before we pose the uniqueness question, we need to consider the role of symmetry. Let (F, (r be a p.c.f.s.s,set, and let T/be a symmetry group of F.

Definition 6.10. C E ~ is 7-t-invariant if for each h E E(foh, goh)=s

f, g E C(V (~

r is 7-t-invariant if rh(i) = ri for all h E ~ . (Here h is the bijection on I associated with h). L e m m a 6.11. (a) Let C = EA. Then E is ~-l-invariazit if and only if: (6.13)

ah(x) h(y)

:

a~y for all x,y E V (~

h E 7~.

(b) Let s and r be ~=invariant. Then AE is 7~-invariant. Proof. (a) This is evident from the equation s

ly) -- -a~y.

85 (b) Let f E C(V(D). Then if h E 7-/,

s

o h , f o h) = Z

r~l ~(f o h o r

ohor

i

I f g C C ( V (~

--~ ~

?,~-1 ~,(f 0 ~)h(i) O h, f o eh(i) o h, )

_- ~

?.;(1i)E(f o ~bh(i), f o ~b~(i)) = ER(f,f).

then writing ~ = A(s

~(goh, goh) ~_s

if fly(o) = g then as f o hlv(o ) = g o h,

oh, f oh)=ER(f,f),

and taking the infimum over :, we deduce that for any h C ~, ~'(goh,goh) _< ~'(g,g). Replacing g by g o h and h by h -1 we see that equality must hold.

[]

If the fractal F has a non-trivial symmetry group ~ ( F ) then it is natural to restrict our attention to ~(F)-symmetric diffusions. We can now pose the uniqueness problem. P r o b l e m 6.12. (Uniqueness). Let (F, (r be a p.c.f.s.s, set, let 7"/be a symmetry group of F, and let r be 7~-invariant. Is there at most one 7g-invariant C E D i such that A(s = ),s (Unless otherwise indicated, when I refer to fixed points for nested fractals, I will assume they are invariant under the symmetry group GR generated by the reflections in hyperplanes bisecting the lines Ix, y], x, y E V(~ The following example shows that uniqueness does not hold in general. E x a m p l e 6.13. (Vicsek sets - see [Me3].) Let (F, r 1 < i < 5) be the Vicsek set - see Section 2. Write {zl, z2, z3, z4,} for the 4 corners of the unit square in R 2. For a,fl,3, > 0 let A(a,f~,~) be the conductance matrix given by a12 : a23 : a34 ---- a41 = or,

ot13 = ~ ,

a24 = - y ,

where aij : az~ z~. If 7t is the group on F generated by reflections in the lines [Zl, z3] and [z2,z4] then A is clearly 7~-invariant. Define &,/~, ~ by

A(EA)= CA(W,~, 7)" Then several minutes calculation with equivalent networks shows that

(6.14)

~ =

a ( a + 13)(a + 7) 5a 2 + 3a/3 + 3a'y + / ~ 7 '

5(" + ~) -

~,

7= 89 If (1,13,3,) is a fixed point then ( ~ , ~ , ~ ) -- (0,0~,0~,) for some 0 ~ 0, so that S o ~ - I - 5 , and this implies that f17 = 1. We therefore have that (1, fl,f~-l) is a fixed point (with A -- 89 for any f~ E (0, oo) Thus for the group 7/ uniqueness does not hold.

=~,~=~7.

86 However if we replace 7-/by the group ~n = G(F), generated by all the symmetries of the square then for EA to be QR-invariant we have to have/3 = 7. So in this case we obtain (6.15)

~ ( . + f~)2

~(a, fl) -- 5a 2 + 6~fl + ~ 2 ,

~(~,~)

+~)

~.

This has fixed points (0,~),/3 > 0, and (a, a), a > 0. The first are degenerate, the second not, so in this ease, as we already saw in Section 2, uniqueness does hold for Problem 6.12. This example also shows that A is in general non-linear. As these examples suggest, the general problem of existence and uniqueness is quite hard. For all but the simplest fractals, explicit calculation of the renormalization map A is too lengthy to be possible without computer assistance - at least for 20th century mathematicians. LindstrCm ILl] proved the existence of a fixed point E E I) si for nested fractals, but did not treat the question of uniqueness. After the appearance of [L1], the uniqueness of a fixed point for LindstrCm's canonical example, the snowflake (Example 5.26) remained open for a few years, until Green [Gre] and Yokai [Yo] proved uniqueness by computer calculations. The following analytic approach to the uniqueness problem, using the theory of quadratic forms, has been developed by Metz and Sabot - see [Me2-Me5, Sabl, Sab2]. Let l~+ be set of symmetric bilinear forms Q(f, g) on C(V (~ which satisfy

Q(1,1) = 0, Q ( f , / ) _> 0 for all f 9

C(V(~

For Q1, Q2 9 l~+ we write Q1 _> Q2, if Q2 - Q1 9 • + or equivalently if Then I3 C l~+; it turns out that we need to consider the action of A on 1~+, and not just on 13. For Q 9 l ~ + , the replication operation is defined exactly as in (6.2)

Q2(f,f) >_Ql(f,f) for all f 9 C(V(~

M

(6.16)

Qn(f,g)=Er~lQ(for162

f,g 9

i=1

The decimation operation is also easy to extend to IVY+ :

T(Qn)(g,g) = inf{QR(f,f): f 9 C(V(~ we can write

= g};

T(Q R) in matrix terms as in (4.24). We set A(Q) = T(Qn).

L e m m a 6.14. The m a p A on M+ satisfies: (a) A : M + ~ ~+, and is continuous on int(M+). (b) A(Q1 + Q2) _> A(Q1) + A(Q2). (c) A(OQ) = 0A(Q)

Proof. (a) is clear from the formulation of the trace operation in matrix terms.

87

Since the replication operation is linear, we clearly have QR = (0Q)R = ~QR. (c) is therefore evident, while for (b), ifg ~ C(V(~

+ Qf,

T(QR)(g,g) = inf{Q~ ( f , f ) + Q~ ( f , f ) : ftv(o) -:- g} > i n f { Q ~ ( f , f ) : fly(o) = g} +inf{Q2R(f,f): fly(o) = g} = T(Q1R)(g,g) + T(QR)(g,g).

[]

Note that for 8 E D i, we have E(f, f ) = 0 if only if f is constant. D e f i n i t i o n 6.15. For 81,82 C ~)i set

= inf{ E1(f, f) constant}. 82(f, f ) : f non Similarly let M(81/82) = sup ~r ~$1(f, f ) : f non constant}. Note that (6.18)

M(~/82)

=

m(82/$~) -~.

L e m m a 6.16. (a) For s

E ]I)i, 0 < m(81,82) < oo . (b) l-f El, & e D*i then m(81/~2) = M(E1/&) ff and only if82 = 481 for some

A>O.

(c) If •,E2,E3

E D i then

m(E1/e ) >_re(elf82) m(82183), M(E1/E3) _< M(81/E2) M(s

Proof. (a) This follows from the fact that Ei are irreducible, and so vanish only on the subspace of constant functions. (b) is immediate from the definition of m and M. (c) We have m(,-qx/Sa) = ii~f 81 (f, f) 82(f, f) > m(81/$2)rn(82/E3); 82(f,f)

E3(f,f)

--

while the second assertion is immediate from (6.18). D e f i n i t i o n 6.17.

[]

Define M(81s dH(81,82)----log m(Eas

81,82 E

Di"

Let pD i be the projective space D i / ~-, where 81 ~ E2 if 81 = 482. dH is called Hilbert's projective metric - see [Nus], [Me4].

88 = 0 if and only if s = )~s for some )~ > O. (b) dH is a pseudo-metric on D ~, and a metric on p D ~. (c) Ifs163163 E E) ~ then for aO,al > O,

P r o p o s i t i o n 6.18. (a) d H ( ~ 1 , s

d~(s ~0 G0 + ~1s

< ma~(dz(s E0), dz(s

In particular open balls in dH are convex. (d) (pD i, dH) is complete. Proof. (a) is evident from Lemma 6.17(b). To prove (b) note that dH(s163 > O, and that dH(s s = dH(s s from (6.18). The triangle inequality is immediate from L e m m a 6.17(c). So dH is a pseudo metric on D i. To see that dH is a metric on p D i, note that m(Ael/e2) = A-~(s163

A > 0,

from which it follows that dH(As s = dH(s s and thus dH is well defined on p/l) i. The remaining properties are now immediate from those of dH on I])i. (c) Replacing s by ('~(s163163 s we can suppose that m(s163 = m(s163 Write Mi = M ( s 1 6 3

= m.

Then if ~" = so Go + a1s

M(s

= inf aos

I

f) + a l s s

f)

~ o~0m(s163 -~- o ~ 1 m ( s Similarly M ( s

= 0~0 "4- 0~1.

~ aoMo + aiM1. Therefore

exp dH(E,~) < (~o/(~o + ~l))(Mo/m) + (~1/(~o + < B(s

m~x(Mo/~,M~/~).

It is immediate that if s E B ( s then dH(s163 is convex. For (d) see [Nus, Thin. 1.2].

Theorem

~1))(M1/~)

6.19. Let s163

+ (1 -- A)s

< r, so that []

E IDi. Then

(6.19)

m(A(s163

> m(s163

(6.20)

M(A(s163

A(Q) + ah(s

89 and since A(Q) _> 0, this implies that A(s - h A ( & ) _> 0. So a < m(A(E1),A(s and thus m(s s 0% > 0}. Clearly we have I~* C D s~. We have the following existence theorem for nested fractals. T h e o r e m 6.23. (See ILl, p. 48]). Let (F, (r Then A has a tlxed point in D*.

be a nested fractal (or an ANF).

Proof. Let s E D*, and let a l , ...ak be the associated 0, Q~, z E V (~ be the continuous time Markov chain (Y,~,n > 0, Q ~ , z E V (~ be the discrete time skeleton L ~~ ~~ 0 0 ) ' " ' " ' ~~(k) be the equivalence classes of edges 0 a i if {x, y} E E~ i). Then if {x, y} E E~ j),

conductivities. Let (Yt, t > associated with CA, and let of Y. in (V (~ ) Eo), so that A~ v

ai Q'(Y1 = V) - ~ y # ~ A~v" As cl = ~ A~ v does not depend on x (by the s y m m e t r y of V (~ the transition vgz probabilities of Y are proportional to the a i. Now let R ( A ) be the conductivity matrix on V (1) attained by replication of A. Let (Xt, t >_ O, I?~, x e V (1)) and (2~,n > 0, P~,z c VO)) be the associated Markov Chains. Let To,T1, ... be successive disjoint hits (see Definition 2.14) on V (~ by -~,~. Write .4 = A(A), and ~ for the edge conductivities given by A. Using the trace theorem, IP~(2T~ -= y) = ~ j / c l if {x, y} 9 E~ i). Now let x l , y l , y 2 (6.23)

9 V (~ with Ix - Yll < ]x - y2]. We will prove that ]~z'(2T~ = Y2) < IP*'(2T~ = Yl).

Let H be the hyperplane bisecting [Yl,Y2], let g be reflection in H , and x2 = g(xl). Let T = min{n > 0: .~,~ 9 V (~ - {xl}},

91

so that T1 = T F~l-almost surely. Set

/~(z) = E ~ i(r O, f,,(x) + f,~(g(x)) >_ O,

(6.25a) (6.25b)

x E J12, x E J12.

Since f0 = 1~ - ly2, a n d y l E J12, f0 satisfies (6.25). Let x E A c U J 1 2 and suppose fn satisfies (6.25). If p(x,y) > 0, and y C J~2, then x , y are in the same 1-cell so if y' = g(y), y' is also in the same 1-cell as xl and Ix - y'] < Ix - y]. So (since $A 6 •*), p(x,y') > p(x,y) and using (6.25b), as f,~(y') >_ O,

p(~, y)/n(y) + p(~, y')/,~(y')>_ p(~, y)(/,,(y)+ .:,,(g(y))_> o. Then by (6.24), fn+l(x) > 0. A similar argument implies that fn+l satisfies (6.25b). So (f,~) satisfies (6.25) for all n, and hence its limit f ~ does. Thus foc(xl) = I?~(-~T = Yl) -- ~(XT = Y2) > 0, proving (6.23). From (6.23) we deduce that ~l > ~2 > ... > ~k, so that A : D* ~ D*. As A'(Oa) = 0A'(a), we can restrict the action of A' to the set {.

> ... >

>

0, y :

=

This is a closed convex set, so by the Brouwer fixed point theorem, A ~ has a fixed point in ]D*. [] R e m a r k 6.24. The proof here is essentially the same as that in LindstrOm ILl]. The essential idea is a kind of reflection argument, to show that transitions along shorter edges are more probable. This probabilistic argument yields (so far) a stronger existence theorem for nested fractals than the analytic arguments used by Sabot [Sabl] and Metz [Me7]. However, the latter methods are more widely applicable. It does not seem easy to relax any of the conditions on ANFs without losing some link in the proof of Theorem 6.23. This proof used in an essential fashion not only the fact that V (~ has a very large symmetry group, but also the Euclidean embedding of V (~ and V (1). The following uniqueness theorem for nested fractals was proved by Sabot [Sabl]. It is a corollary of a more general theorem which gives, for p.c.f.s.s, sets, sufficient conditions for existence and uniqueness of fixed points. A simpler proof of this result has also recently been obtained by Peirone [Pe].

92 be a nested fractal. Then A has a unique ~R-invariant

T h e o r e m 6.25. Let (F, (r non-degenerate fixed point.

Definition 6.26. Let s be a fixed point of A. The resistance scaling factor of C is the unique p > 0 such that

A(E) = p-1 E. Very often we will also call p the resistance scaling factor of F : in view of Corollary 6.21, p will have the same value for any two non-degenerate fixed points.

Proposition 6.27. Let (F, (r

be a p.c.s set, let (r,) be a resistance vector, and let s be a non-degenerate fixed point of A. Then for each s E {1, ...M} such that r(~) C V (~ (6.27)

r~ p-1 < 1.

Proof. Fix 1 < s < M, l e t x = 7r(~), and let f = 1~ E C(V(~ EA(f,f) =

E A~v = IA~I. yE V (~ y#x

=- p-lEA,

Let g : lx C C(V(1)). As A(s (6.28)

Then

p-l[A~z[ = A ( E A ) ( f , f ) < EAR(g,g) :

since g is not harmonic with respect to CA R, strict inequality holds in (6.28). By Proposition 5.24(c), x is in exactly one 1-complex. So

SAR(g,g) = ~

r:-lSA(g o r

or

= r:llAzzl,

i and combining this with (6.28) gives (6.27).

[]

Since r~ = 1 for nested fractals, we deduce C o r o l l a r y 6.28. Let (F, (r

be a nested fractal. Then p > 1.

For nested fractals, many properties of the process can he summarized in terms of certain scaling factors. D e f i n i t i o n 6.29. Let (F, (r be a nested fractal, and E be the (unique) nondegenerate fixed point. See Definition 5.22 for the length and mass scale factors L and M. The resistance scale factor p of F is the resistance scaling factor of E. Let (6.29

~ = Mp ;

we call T the time scaling factor. (In view of the connection between resistances and crossing times given in Theorem 4.27, it is not surprising that ~- should have a connection with the space-time scaling of processes on F.) It may be helpful at this point to draw a rough distinction between two kinds of structure associated with the nested fractal ( F , r The quantities introduced in Section 5, such as L, M, the geodesic metric dF, the chemical exponent 7 and the dimension dw(F) are all geometric- that is, they can be determined entirely by a geometric inspection of F. On the other hand, the resistance and time scaling

93 factors p and ~- are analytic or physical- they appear in some sense to lie deeper than the geometric quantities, and arise from the solution to some kind of equation on the space. On the Sierpinski gasket, for example, while one obtains L = 7 = 2, and M = 3 almost immediately, a brief calculation (Lemma 2.16) is needed to obtain p. For more complicated sets, such as some of the examples given in Section 5, the calculation of p would be very lengthy. Unfortunately, while the distinction between these two kinds of constant arises clearly in practice, it does not seem easy to make it precise. Indeed, Corollary 6.20 shows that the geometry does in fact determine p: it is not possible to have one nested fractal (a geometric object) with two distinct analytic structures which both satisfy the s y m m e t r y and scale invariance conditions. We have the following general inequalities for the scaling factors.

Proposition 6.30.

Let (F, (r

be a nested fractal with scaling factors L, M, p, T.

Then (6.30)

L>I,

M_>2,

M>L,

T=Mp>L

2.

Proof. L > 1, M > 2 follow from the definition of nested fractals. If0 = diam(V(~ then, as VO) consists of M copies of V (~ each of diameter L -10, by the connectivity axiom we deduce M L - I O > O. Thus M > L. To prove the final inequality in (6.30) we use the same strategy as in Proposition 6.27, but with a better choice of minimizing function. Let 7-/be the set of functions f of the form f ( x ) = O x + a , where x C L~d and O is an orthogonal matrix. Set 7-/,~ = {fly(~), f E 7-/}. Let 0 = sup{C(f, f ) : f E 7-/0}: clearly 8 < ~ . Choose f to attain the supremum, and let g E 7-/ be such that f = gig(o). Then if fx = gig(l) M

p-10 = p-lE(f, f) = h(E)(f, f) < ER(g~,gl) = ~

E(gl o r

or

i=1

However, gl o r is the restriction to V (~ of a function of the form L - l O x + ai, and so E(g o r o r Roe(f, f),

f e

c(v(~)).

94

So AI(E) >_ A0(E) for any s 9 D. I f m =

p11s = AI(s

_> A l ( m s

m(s163

_> Ao(ms

then =

mPols

>_poIs , []

which implies that Po _> Pl.

7. D i f f u s i o n s

on p.c.f.s.s, sets.

Let (F, (r be a p.c.f.s.s, set, a n d r, be a resistance vector. We assume that the graph (V (1), E l ) is connected. Suppose that the renormalization map A has a non-degenerate fixed point s (~ = CA, so that A(s (~ : p-1s176 Fixing F, r, and s in this section we will construct a diffusion X on F, as a limit of processes on the graphical approximations V ('0. In Section 2 this was done probabilistically for the Sierpinski gasket, but here we will use Dirichlet form methods, following [Kus2, Ful, Ki2]. D e f i n i t i o n 7.1. For f 9 C(V(n)), set (7.1)

E(n)(f,f):p

n ~ r=lC(~ wEw~

This is the Dirichlet form on V (n) obtained by replication of scaled copies of s (~ where the scaling associated with the map Cw is p'~r~1. These Dirichlet forms have the following nesting property. 7.2. (a) For n > 1, Tr(s ('~-D : s (b) Is 9 C(V(")), and g = flv(--~) then $(")(f, f) _> g("-l)(g,g). (c) s is non-degenerate.

Proposition

Proof. (a) Let f 9 C(V(~)). Then decomposing w 9 W~ into v. i, v 9 W ~ - l , (T.2)

E(n)(f,f)=P n ~

r:l~'~r/1E(~176162176162176162176

vEW~_~ = pn--I

i

r:lE(1)(f~'f=)'

E yEW,,_ ~

where f~ = f o Cv E C(VO)). Now let g E C(V('~-I)). fv[v(o) = g o r = g,. As s (~ is a fixed point of A, (7.3)

inf{s

=g~}

=pinf{Rs176 =

=

Summing over v E W,~-I we deduce therefore

v

If

flv(.-l) = g

then

95 For each v C W n - 1 , let h~ E C(V (U) be chosen to a t t a i n the infimum in (7.3). We wish to define f E C(V('O) such t h a t (7.4)

f or

-- hv,

v C W~-I.

Let v E W n - 1 . We define f(r

= h~(y),

Y E V(1).

We need to check f is well-defined; b u t if v, u are distinct elements of W,~-I and x = r = r t h e n x C V (n-D by L e m m a 5.18, and so y, z E V (~ Therefore f(r

= h~(v) = g~(y) = g ( x ) = f ( r

so t h e definitions of f at x agree. (This is where we use the fact t h a t F is finitely ramified: it allows us to minimize separately over each set of the form V(1)). So E(n)(f, f ) = ~ ( n - 1 ) ( g , g ) , a n d therefore Tr (s = g'('~-U. (b) is evident from (a). (c) We prove this by induction. E (~ is non-degenerate by hypothesis. s is non-degenerate, a n d t h a t E ( " ) ( I , f ) = 0. From (7.2) we have

E(")(f,f) = p ~

r:~E("-~)(f o r

Suppose

f or

vEWx

a n d so f o r is constant for each v E W1. Thus f is constant on each I-complex, a n d as (V (1), E1 ) is connected this implies t h a t f is constant. [] To avoid clumsy n o t a t i o n we will identify functions with their restrictions, so, for example, if f E C(V(~')), a n d m < n, we will write s i n s t e a d of

E(m) (flv(~), flv(~)). D e f i n i t i o n 7.3. Set V (~ = U~=0 V(~). Let U -- { f : V (~) --* R}. Note t h a t the sequence (C(n)(f, f))~=1 is non-decreasing. Define 7)' ----{f e U : supC(n)(f, f ) < (x)}, n

s

= sups

f,g C 7)'.

E ~ is the initial version of the Dirichlet form we are constructing. Lemma

7.4. s is a s y m m e t r i c

Markov

form on 7)~.

Proof. s clearly inherits the properties of symmetry, bilinearity, and positivity from the s If f E 7)', a n d g = (0 V f ) A 1 then s n, f E If,

E(m)(f'f)= E

P'~r~'IE(m-n)(f~162176162

wEW~

Letting m --* oo it follows, first that f o r

E Z)~, and then that (7.5) holds.

[]

If H is a set, and f : H ---, ]~, we write (7.6)

O s c ( f , B ) = sup I f ( x ) - f(Y)l,

B C H.

~,y6B

Lemma

7.6. There exists a constant co, depending only on $, such that

f E C(V(~

_< coE(~

O s c ( f , V (~

Proof. Let /~0 = {{x,y} : A,y > 0}. As s is non-degenerate, (V(~ is connected; let N be the maximum distance between points in this graph. Set a = m i n { A , y , { x , y } e E0}. If z, y e Y (~ there exists a chain x = x o , x l , . . . , x n = y connecting x, y with n < N, and therefore,

If(z) - f(y)l 2 _
P0, t h e n as r7 > P0 ~ p(rT), we have an example of an affine nested fractal with a non-regular fixed point. From now on we take s to be a regular fixed point. (See [Kum3] for the general situation). W r i t e 7 = m a x / r i / p < 1. For x, y 9 F , set w(x,y) to be the longest word w such t h a t x, y 9 F ~ . Proposition

7.10.

(Sobolev inequality). Let f 9 T~'. Then if s

is a regu/ar

fixed point (7.8)

i f ( x ) _ f(y)]2 ~ c2r~(,,y)p-l~(z,y)ls

x,y 9 V (~176

Proof. Let x, y 9 Y(,'), let w = w(z,y) and let Iwl = m. We prove (7.8) by a s t a n d a r d kind of chaining argument, similar to those used in continuity results such as Kolmogorov's lemma. (But this argument is deterministic and easier). We m a y assume n > m. Let u 9 W," be an extension of w, such t h a t x 9 V(~ such a u certainly exists, as x 9 V~( ~ Write uk = ulk for m < k < n. Now choose a sequence zk, m < k < n such t h a t z," = x, and zk 9 V~(~ for k < m < n k 9 { m , . . . , n - 1} we have zk, zk+l 9 V(~ ). So

(7.9)

If(z,,)- f(z.~)l < ~

lf(zk+1) - f(zk)l

n--1

< Z le,=m

(e:~

I/"

1. For each

98

n--1 Tuk ^_k+m~l/2 k=m As s is a regular fixed point, 7 = m a x i ri/p < 1, so the final sum in (7.9) is b o u n d e d CO by ( ~ k = m 7 k - m ) U2 = c3 < co. Thus we have

If(x) - f(zm)l ~ < c l c : ~ p - " E ' ( f , f), a n d as a similar b o u n d holds for If(y) - f(zm)[ 2, this proves (7.8).

[]

We have not so far needed a measure on F . However, to define a Dirichlet form we need some L 2 space in which the domain of ~ is closed. Let # be a p r o b a b i l i t y measure on (F, B(F)) which charges every set of the form F ~ , w E Wn. L a t e r we will take # to be the Bernouilli measure P0 associated with a vector of weights 8 E (0, ~ ) M , b u t for now any measure satisfying the condition above will suffice. As # ( F ) = 1, C(F) C L 2 ( F , # ) . Set

= {f e C ( F ) :

e(f,f) Proposition

=

flv(~) 9 :D'}

e'(flv(~),flv(~)),

f 9 :V.

7.11. (~, :D) is a dosed symmetric form on LZ(F, p).

Proof. Note first t h a t the condition on p implies t h a t if f , g 9 l) then [[f - g [ [ 2 = 0 implies t h a t f = g. We need to prove t h a t V is complete in the norm ][f[[~, -e ( f , f ) + Ilfll]. So suppose (fn) is Cauchy in [[. lie,- Since ( f , ) is Cauchy in H" [Is, passing to a subsequence there exists f 9 L 2 ( F , # ) such t h a t f,~ ~ f #-a.e. F i x x0 9 F such t h a t fn(xo) --~ f(x). T h e n since fn - f m is continuous, (7.8) extends to an e s t i m a t e on the whole of F and so If,,(x) - f,~(x)[ _< l(f,, - f,,,)(x) - (f,, - f,,,)(x0)l + I(f,, - fm)(x0)l I12 ,-,:. I, E(N)(f~ -- f, fn -- f) (_ limoo E(f,~ -- fm, f,~ -- f.',). So E(f,~ - f, fn - f) --~ 0 as n ---* oo, and thus Ill - f.II~,

--+ o.

[]

To show t h a t (C, Z~) is a Dirichlet form, it remains to show t h a t 2) is dense in

L2(F, p). We do this by s t u d y i n g the harmonic extension of a function.

99 D e f i n i t i o n 7.12. Let f e C(V(')). Recall that C('~)(f,f) -~ inf{E("+l)(g,9) : giv(-) -- f}- Let H n + i f E C(V (~+D) be the (unique, as E (n+l)) is non-degenerate) function which attains the infimum. For x G V (~) set

H~f(x) = lim H , ~ H , ~ - I . . . H , ~ + l f ( X ) ; note that (as H ~ + l f -- f on V (~)) this limit is ultimately constant. 7.13. Let g be a regular fixed point. (a) H,~f has a continuous extension to a function H~f C 7) M C(F), which satisfies

Proposition

~(H,J, Hnf) = E(n)(f, f).

(b)

f, a c C(F) E(H~f,g) -- g('~)(f,g).

(7.10)

Proof. From the definition of H,~+I, ~(n+i)(Hn+lf, H ~ + l f ) -- s

Thus

E(m)(H,~f,H~f) = C(~)(f, f ) for any m, so that H ~ f E 7)~ and

E(Hr, f, Hnf) = C(n)(f,f),

f E C(V(")).

If w E Win, and x, y E V (~) M F,o then by Proposition 7.10 (7.11)

] H u f ( x ) - H , I ( v ) ] 2 0

(7.12)

E(/'g)=

E P%'~lE(f~162176162 wGW~

(b) Fo~, / 9 79, (7.13) (7.14)

(7.15)

If(x) -/(y)l 2


0) be the resolvent

E~(vof, g) = (f,g), if U~ has a density u~(x,y) with respect to #, then a formal calculation suggests that E~(u~(~,.),9)

= E~(u~=,9)

= (~=,g) = g(~).

We can use this to obtain the existence and continuity of the resolvent density u~. (See [FOT, p. 73]).

103

T h e o r e m 7.20.

(a) For each x 9 F there exists u s 9

(7.19)

for all

Ea(u~, f ) = f ( x )

suchthat

f 9

(b) Writing u~,(x, y) = u~(y), we have

u~(=,y) = ,~.(y,=)

:or a11 =,y e F.

(c) us(.,-) is continuous on F • F and in particular lu~(=,y) - ~ ( = , y ' ) l 2 < R(y,y')~o(=,=).

(7.20)

(d) ~.(=,y) is the resolvent density ~or X: ~or f e C ( f ) , E~

:

e-~tf(Xt)dt = U~f(z) =

/

u~(z,y)f(y)~(dy).

(e) There exists c2(a) such that (7.21)

~ ( ~ , y ) O) such that for all bounded

measurable f l(x.)es

=

~)Lt.(d~),

a.s.

Pro@ These follow from the estimates on the resolvent density u~. As u~ is bounded and continuous, we have that x is regular for {x}. Thus X has jointly measurable local times (L~,x E F , t > 0). Since X is a symmetric Markov process, by Theorem 8.6 of [MR], L~ is jointly continuous in (x, t) if and only if the Gaussian process Yx, x E F with covariance function given by EYaYb = ul(a,b)~

a,b E F

is continuous. Necessary and sufficient conditions for continuity of Gaussian processes are known (see [Tall), but here a simple sufficient condition in terms of metric entropy is enough. We have

E(Ya - yb)2 = ul(a,a) - 2ul(a,b) + ul(b,b) < clR(a,b) U2. Set r(a, b) = R(a, b) 1/2 : r is a metric on F. Write N~(e) for the smallest number of sets of r-diameter e needed to cover F. By (7.17) we have R(a, b) < c~/'~ if a, b e Fw and w E W~. So N~(ct~/n/2) < #W,~ = M ~, and it follows that N~(c) < e2c - ~ ,

where fl = 2 log M~ log ~-1. So

fo + (lOg~V~(~))~/~a~ < ~,

105

and thus by [Du, Thm. 2.1] Y is continuous.

[]

We can use the continuity of the local time of X to give a simple proof that X is the limit of a natural sequence of approximating continuous time Markov chains. For simplicity we take # to be a Bernouilli measure of the form It -- It0, where 0i > 0. Let It,, be the measure on V ('0 given in (5.21). Set

A'~ = ./~ L~It,~(dx), "r: = inf{s : A~ > t}, x;' = x v.

(a) (X?,t >_ 0 , I ? ' , x 6 V (~)) is the symmetric Markov process associated with $ ('~) and L2(V ("), It,). Theorem

7.22.

(b) X~ --+ Xt a.s. and uniformly on compacts. Proof. (a) By Theorem 7.21(a) points are non-polar for X. So by the trace theorem (Theorem 4.17) X n is the Markov process associated with the trace of $ on L 2 ( V ( ' 0 , I t , ) . But for f 6 :D, by the definition of s Tr

(elv ( - ) ) ( f , :) =

e(n)

(flv(-),fl,:~-)).

(b) As F is compact, for each T > 0, (LT, 0 < t < T, x 6 F ) is uniformly continuous. So, using (5.22), if T2 < T1 < T then A t ~ t uniformly in [0,T1], and so T~ --+ t uniformly on [0, T2]. As X is continuous, X~ ~ X uniformly in [0,T2]. [] R e m a r k 7.23. As in Example 4.21, it is easy to describe the generator Ln of X ~. Let a('O(x, y), x, y G V('*) be the conductivity matrix such that 1

E(")(f,:) = ~ E a(n)(~,V)(f(~)- f(V)) ~ z,y

Then by (7.1) we have (7.23)

a('q(x,y) ---

E

l(,,yev!0))p n r~--1A ( r

--1 ( x ) , r w --1 (y)) ,

w6W~

where A is such that E (~ -- CA, and A(x,y) = A=y. Then for f E L2(V('q,#,~), (7.24)

Lnf(x) -- # n ( { x } ) - '

E

a(n)(x'Y)( f ( y ) - f(x)).

yEV(~)

Of course T h e o r e m 7.22 implies that if (Y'~) is a sequence of continuous time Markov chains, with generators given by (7.24), then Y'~ TM ~X in D([0, oo), F).

106 8. Transition D e n s i t y Estimates. In this section we fix a connected p.c.f.s.s, set (F, ( r a resistance vector ri, and a non-degenerate regular fixed point ~A of the renormalization m a p A. Let P = P0 be a measure on F , and let X -- ( X t , t > O, IP~,z C F ) be the diffusion process c o n s t r u c t e d in Section 7. We investigate the transition densities of the process X : initially in fairly great generality, but as the section proceeds, I will restrict t h e class of fractals. We begin by fixing the vector 0 which assigns mass to the 1-complexes r in a fashion which relates # e ( r with ri. Let/3i = r l p - l : by (7.8) we have

fli < l,

(8.1)

l < i < M.

Let a > 0 be t h e unique positive real such t h a t M

(8.2)

ZZ?

-- 1.

i=I

Set

(8.3)

Oi =

fiT,

1
0, i = 1 , . . . , d} be a d-dimensional Brownian motion. In this case the Gaussian space is obtained by taking H ---- L2(]R+;]Rd), and for any h C H the variable W(h) is the Wiener stochastic integral ~i=ld f ~ h~dW~. 2. Suppose that (T, B, #) is a measure space such that B is countably generated, and {W(A), A 9 B, #(A) < oc} is a family of random variables such that each W(A) has the distribution g ( 0 , #(A)), W(A U B) = W(A) + W(B) if A and B are disjoint, and { W ( A 1 ) , . . . , W(An)} are independent whenever the sets { A 1 , . . . , An} are pairwisedisjoint. In this case the Gaussian space is obtained by taking H = L2(T, B, p), and W(h) is given by the stochastic integral

W(h) = IT h(t)W(dt). We will say that W is a white noise on (T, B, #). Note that Example 1 is a particular case of example 2 by setting T = IR+ • { 1 , . . . , d} and tt equals to the product of the Lebesgue measure times the uniform measure on { 1 , . . . , d}.

127

3. Let v be a zero mean Gaussian measure on a real separable Banach space 113 with full support. Consider the inclusion j : II3" --* L2(v) which is linear, continuous and one-to-one. Set 7-ll = j(IB*). Then 7-ll is a Gaussian subspace of L~(v).

Remarks: 1. Consider the case of a d-dimensional Brownian motion W defined on the canonical probability space f~ = C0(N+, IRa). The subspace H 1 of f~ formed by the functions of the form ~o(t) = fg r ~b E H = L2(IR+; IRa) is called the Cameron-Martin space. Equipped with the scalar product H 1 = "

H 1 is isometric to H, and the inclusion i : H 1 -+ f~ is continuous. 2. In the context of Example 3, one can show (eft Gross [34], Kuo [55]) that the dual map j* of j is a compact operator from 7-/1 into IB with a dense image. Then the image H 1 := j*(~x) is a subspace of 113, and the triple (IB, H 1, u) is called an abstract Wiener space. When t13 = Co(N+, IRd), and u is the Wiener measure, then H 1 is the Cameron-Martin space. In fact, given a point measure m = Chto E 113", c E IRd to >_ 0, clearly j ( m ) = c . W(to) and j*j(m)(t) = c(to A t) because

j*j(m)~(t)

= (j*j(m),e~ht)~,~. = (j(m),j(e~t))L2(,) = E ( ( c . W(to)(e,. W(t)) = c,(to A t),

where e, is the ith vector of the canonical basis of IRa. Notice that the function ~(t) = c(to A t) is absolutely continuous and its derivative has an L2-norm equals to v~llcll which is the L2-norm of j ( m ) in L2(v). As a consequence, j*(7-(1) consists of all absolutely continuous functions ~ : IR+ ~ IRd which vanish at zero at have a square integrable density, that is, j*(7-/1) is the Cameron-Martin space. Let us first introduce the derivative operator D. We will follow an approach analogous to the definition of the Sobolev spaces in finite dimensions. We denote by C ~ ( I R n) (resp. C~(IRn)) the set of all infinitely differentiable functions f : IR'~ IR such that f and all of its partial derivatives have polynomial growth (resp. are bounded). Let S (resp. Sb) denote the class of random variables of the form

F = f ( W ( h l ) , . . . , W(hn)),

(1.1)

where f belongs to Cp(IR n) (resp. C~(IR'~)), h i , . . . , h,~ are in H, and n > 1. These random variables are called smooth. We will denote by P the subset of S formed by random variables of the form (1.1) where f is a polynomial. If F has the form (1.1) we define its derivative D F as the H-valued random variable given by

O F = f i ~fx ( W ( h l ) , . . . , W ( h n ) ) h , .

(1.2)

4=1

Notice that for any element h E H the scalar product (DF, h}n coincides with the directional derivative ~a ~,~hl1~=0,where F~h is the shifted random variable

F `h = f ( W ( h l ) + c(h, h , ) H , . . . , W(h~) + c(h, hn)n).

128

In order to show that the operator D is closable we need the following integrationby-parts formula:

1.1.1 Suppose that F is a smooth random variable and h E H. Then

Lemma

E((DF, h)g) = E(FW(h)).

(1.3)

Proof: We can assume t h a t the norm of h is one. There exist orthonormal elements of H, e l , . . . , en, such t h a t h = el and F is a smooth random variable of the form F = f ( W ( e l ) , . . . , W(e,)). Let r

denote the density of the s t a n d a r d normal distribution on IRn. Then we have

E(H + FGW(h)).

(1.4)

As a consequence of the above lemma, D is closable as an operator from DO(f/) to DO(f~; H) for any p _> 1. In fact, let {FN, N >_ 1} be a sequence of smooth random variables such t h a t FN converges to zero in D'(ft) and the sequence of derivatives DFN converges to r / i n LP(f/; H). Then, from Lemma 1.1.2 it follows t h a t r / i s equal to zero. Indeed, for any h E H and for any smooth random variable F C Sb such t h a t FW(h) is bounded, we have

E((~, h)HF) =- limE((DFN, h)HF) N

= limE(-FN(DF, h)H + FNFW(h)) = O, N

because FN converges to zero in D ~ as N tends to infinity, and the random variables (DF, h)H and FW(h) are bounded. This implies r / = 0. We will denote the domain of D in LP(t2) by ID I'p, meaning t h a t 1DI'p is the closure of the class of smooth r a n d o m variables S with respect to the norm IIFIh,, = [E(IFI" ) + E(IIDF[[;)][. For p = 2, the space ]131'2 is a Hilbert space with the scalar product (F, G)l,2

:

E(FG) + E((DF, DG)H).

We can define the iteration of the operator D in such a way that for a smooth random variable F, the derivative DkF is a random variable with values on H |

129

Then for every p > 1 and any natural number k we introduce a seminorm on S defined by k

IIFIl[,p : E(IFI p) + ~ E(IIDJFII~|

(1.5)

j=l

As in the case k = 1 one can show t h a t the operator D k is closable from S C LP(ft) into LP(f~;H| p >_ 1. For any r e a l p _> 1 and any natural number k _> 0, we will denote by 119k'p the completion of the family of smooth random variables S with respect to the norm I1 9 IIk,~. Note t h a t IDj'p C I19k'q if j _> k and p >_ q. Let V be a real separable Hilbert space. We can also introduce the corresponding Sobolev spaces of V-valued r a n d o m variables. More precisely, S v will denote the family of V-valued smooth r a n d o m variables of the form

F = f i F~b,

b ~ V,

Fj ~ S.

j=l

We define D k F = ~j~=l DkFj | b , k _> 1. Then D k is a closable operator from SV C LP(~; V) into LP(f~; H ~k | V) for any p >_ 1. For any integer k > 1 and any real number p _> 1 we can define the seminorm on S v by k P IIFIIk,p,v = E(IIFII~/) + ~ E(IIDJFII~|174

9

j=l

We denote by IDk'P(V) the completion of Sv with respect to the norm k = 0 we put IlFIIo,~,v

=

p

1

[E(llfllv)]~, and ID~

II 9 Ilk,p,V. For

= LP(a; V).

Consider the intersection

ID~176 = n,>_~ n,, IDk"(V). Then IDa(V) is a complete, countably normed, metric space. We will write ID~176 = ID~. For every integer k _> 1 and any real number p _> 1 the operator D is continuous from IDk'P(V) into IDk-I'P(H | V). Consequently, D is a continuous linear operator from ID~176 into I D ~ ( H | V). Moreover, if F and G are r a n d o m variables in ID~ , then the scalar product (DF, DG)H is also in I19~176 The following result is the chain rule, which can be easily proved by approximating the random variable F by smooth r a n d o m variables, and the function ~ by (~ * ~ ) , where ~ is an approximation of the identity. 1.1.1 Let ~ : IRm ~ IR be a continuously differentiable function with bounded partial derivatives, and fix p >_ 1. Suppose that F = ( F I , . .. , F m) is a random vector whose components belong to the space ID I'p. Then ~(F) E I19_ 0, r = 1 and its support is included in the interval [-1, 1]. Define the function r = r for all e > 0. Set r

=

f

r oo

By the chain rule r belongs to ]1)1'1 and Dr H-valued random variable of the form

= r

Let u be a smooth

n

u = E F,~j, j=l

where Fj C Sb and hj C H. Observe that the duality relation (1.7) holds for F in ID1'1 n L ~ ( ~ ) and for an element u of this type. Note that the class of such processes u is total in L l ( a ; H) in the sense that if v C L l ( a ; H) satisfies E({v, U}H) = 0 for all u in the class, then v - 0. Then we have

IE (r

= =

IE ((D (r IE (r

_< ellr

).

Letting e l 0, we obtain

E ( I { F : o } { D F , U}H) = 0 , which implies the desired result. Let us now state and prove the local property of the divergence.

[]

132

1.3.2 Let u E IDI'2(H) and A C ~ , such that u(w) = O, P a.e. on A. Then 5(u) = 0 a.e. on A.

Proposition

Proof."

Let F be a smooth random variable of the form

F = f(W(h,),...,

W(h~)),

with f E C~~ n) ( f is an infinitely differentiable function with compact support). We want to show t h a t 5(U)I{IMIH=0} = 0, a.s. Consider a function r : IR ~ IR such as that in the proof of Proposition 1.3.1. It is easy to show t h a t the product FCdll~llS) belongs to ~ 1 , ~ Then by the duality relation (1.7) we obtain

Z (5(u)r

= E ({u,D[Fr :

E

D F > . ) + 2E

We claim t h a t the above expression converges to zero as e tends to zero. In fact, first observe t h a t the r a n d o m variables

v~ = Cdllull~)(~, DF)H

+

2Fr

t

2

D~u)u

converge a.s. to zero as e ~ 0, since [[U[[H = 0 implies ~ = 0. Second, we can apply the Lebesgue d o m i n a t e d convergence theorem because we have

Ir162

DF}H]

IIOlI~IlulIHIIDFIIH, 1, the set of V-valued random variables F such t h a t there exists a sequence {(f~n, F~), n _> 1} C ~- x IDI'p(V) with the following properties: (i) f~n T ft , a.s. (ii) F = F~ a.s. on f~n. We then say t h a t ( f ~ , F~) localizes F in IDI'P(V), and D F is defined without ambiguity by D F = DF~ on f ~ , n > 1. The spaces ID~o~(V) can be introduced analogously. Then, if u C ]D1,2(H~ lor j, the divergence ~(u) is defined as a random variable determined by the conditions 5(u)ln ~ = 6(un)la ~ where (fin, un) is a localizing sequence for u.

for

all

n _ 1,

133

1.4

W i e n e r chaos expansions

We will denote by {H~, n E IN} the sequence of the Hermite polynomials defined from the series expansion t2 exp(tx - ~ ) = ~ tnHn(x). (1.13) n=0

Then { v ~ . H ~ , n E IN} is a complete orthonormal system in L2(IR,#) where # is the normal distribution N(0, 1). We will denote by A the set of all sequences a = (al, a2,...), a~ e IN, such that lal = el + a2 + ' " < oc. Let {ei, i > 1} be a complete orthonormal system in H. For any a E A we set oo

(~a : ~

H Hai(W(ei)) ' i=1

where a! = I]~1 a~!. Then the family {Be, a C A} constitutes an orthonormal basis of L2(~,5 ~, P) (we recall that we assume that the a-field jr is generated by W). For any n > 0 we will denote by ~,, the closed subspace of L2(f~) spanned by {(I)~, a C A, la[ = n}. Then ~ n is called the Wiener chaos of order n and we have the orthogonal decomposition =

~)n=0

n-

We will denote by Jn the orthogonal projection on the nth Wiener chaos 7-/n. The Wiener chaos expansion can be extended to the space L2(~; V) of Hilbert valued random variables: L2(~; V) = Q~=0Hn(V), where Tln(V) = 7-[~ | V. The following result characterizes the domain of the operator D in L 2 in terms of the Wiener chaos expansion. P r o p o s i t i o n 1.4.1 We have E(IIDFII2H) : ~ nlldn(F)ll[ ,

(1.14)

n=l

in the sense that this sum is finite if and only if F E ]D 1'2. Moreover, for all n > 1 we have D ( J n ( F ) ) = J n - I ( D F ) . Proof: It suffices to compute the derivative of a random variable of the form (I)~, using the relationship H'~ = H~-l, which follows immediately from (1.13): oo

D(~a)----~a~.E

fi

Ha~(W(ei))Haj-l(W(ej))eJ"

j=l i=l,i#j

Then D((I)~) 9 7"/n_l(H) if [a I = n, and E(HD(Oa)H2H) =

j=l

H~ ai!(aj - 1)! = lal' i=ljr

Now the proposition follows easily. By iteration we obtain E(IIDkF[I2H|

= ~ n(n - 1 ) . . . (n - k + 1)llJn(F)ll~, n=k

[]

(1.15)

134

and F E IDk'2 if and only if ~ - - 1 nkllJ~(F)ll~ < oo. Moreover, for all n > k we have

Dk(Jn(F)) = J~_k(DkF). The next lemma is a consequence of the expression of the operator D in terms of the Wiener chaos. L e m m a 1.4.1 Let G E L2(f~) and q E L2(f~; H) be such that

E(ae(v)) = E((~, ~).), for all v E IDI'2(H). Then G E ID 1'2 and D G = ~. Proof."

We have E((r/, v>t~) = E(Gh(v)) =

E(J~(G)5(v)) = ~ E((D(J~(G)), v)t4), n=l

n=l

hence, Jn-lrl = D(Jn(G)) for each n > 1 and this implies the result.

[]

Remarks: If F is a r a n d o m variable in the space ID 1'1 such t h a t D F = 0, then F = E ( F ) . This is obvious when F E ID 1'2 if we use the Wiener chaos expansion of F and Proposition 1.4.1. In the general case this property can be proved by a duality argument. Indeed, let CN be a function in C~(IR) such that Cy(X) = 9 if Ixl < N, and I~N(x)I _< N + 1. Then one shows t h a t

E(r

- E(F))5(u)) = 0

(1.16)

for any bounded H-valued r a n d o m variable u in the domain of 5. A p p r o x i m a t i n g an a r b i t r a r y random variable u E IDI'2(H) by the sequence of bounded r a n d o m variables {uCk(llull~),k > 1}, where Ck(x) = 1 if Ixl < k, Ck(x) = 0 if Ixl > 2k, and Ir < 2/k, we obtain t h a t (1.16) holds for any u E D I ' 2 ( H ) . Finally, by L e m m a 1.4.1 this implies t h a t C N ( F - E ( F ) ) = E ( r - E ( F ) ) ) , and letting g tend to infinity we deduce the result. As an application of the chain rule and the above p r o p e r t y we can show the following result (see Sekiguchi and Shiota [96]).

L e m m a 1.4.2 Let A E ~ . Then the indicator function of A belongs to ]131'1 if and only if P ( A ) is equal to zero or one. Proof:

Applying the chain rule to a function ~ C C ~ ( ] R ) which is equal to x 2 on [0, 1] yields D I A = D(1A) 2 = 21AD1a.

Consequently, D I A = 0 because from the above equality we get t h a t this derivative is zero on A c and equal to twice its value on A. So, by the previous remark we obtain

1A = P(A).

[]

135

1.5

The white

noise case

Consider the particular case of a white noise {W(A), A 6 I3, #(A) < oo} defined on a measure space (T, B, #). Suppose in addition that the measure # is a-finite and without atoms. In this case (cf. It5 [49]) the multiple stochastic integral of order n provides an isometry between the space L2,T st n , B n , u.n~) of square integrable and symmetric functions of n variables and the nth Wiener chaos ~n. Let us briefly describe how the multiple stochastic integral is constructed. Consider the set g~ of elementary functions of the form k

f(h,...,tn)=

~

air..i lA~lx...xA~(t],...,tn) ,

(1.17)

il,...,in=l

where Ax, A2,..., Ak are pairwise-disjoint sets of finite measure, and the coefficients ai~...i~ are zero if any two of the indices i l , . . . ,in are equal. Then we set k

In(f) =

~

aw..~W(Ail)..'W(A,~).

il,...,in=l

The set gn is dense in L2(Tn), and I~ is a linear map from gn into L2(~) which verifies I~(f~) = I ~ ( s and 0

n](f,O)L2(T~ )

E(I~(f)Iq(g))=

if if

n=/q, n=q,

where f denotes the symmetrization of f. As a consequence, I~ can be extended as a linear and continuous operator from L2(T n) into L2(f~). The image of L2(T n) by In is the nth Wiener chaos 7-/n. This is a consequence of the fact that multiple stochastic integrals of different order are orthogonal, and that In(f) is a polynomial of degree n in W ( A J , . . . , W(Ak) if f has the form (1.17), and those polynomials are included in the sum of the first n chaos. As a consequence, any square integrable random variable F admits an orthogonM expansion in the form F

= E(F) + fi/~(f~),

(1.18)

n=l

where the functions fn E L2(T n) are symmetric and uniquely determined by F. The operators D and 6 can be represented in terms of the Wiener chaos expansion. P r o p o s i t i o n 1.5.1 Let F C ID1'2 be a square integrable random variable with an

expansion of the form (1.18). Then we have O~

DtF : ~ nln-l(fn(',t)).

(1.19)

n=l

Proof: Suppose first t h a t F = L,(f,,), where f,, is a symmetric and elementary function of the form (1.17). Then Dtr= f

~

j = l il,...,in=l

ah...,.W(A~l)...1A,~(t)...W(Ai~)=nln-l(f=(.,t)).

136

Then the results follows easily.

[]

Note that the L2(t2; H)-norm of the right-hand side of (1.19) coincides with expression (1.14). We remark that the derivative DtF is a random field parametrized by T. Suppose that F is a random variable in the space IDN'2, with a Wiener chaos expansion of the form F = End__0In(fn). Then, applying Proposition 1.5.1 N times we obtain that DNF is a random field parametrized by T x given by

D~,..,tNF = ~ n ( n - 1 ) . . - ( n -

N + 1)In-N(f~(',tl,...,tN)).

n=N Note that the L2-norm of this expression is given by (1.15). As a consequence, if F belongs to ID~ = ANID N'2 then fn = ~ E ( D ~F) for every n k 0 (cf. Strooek [100]). We will now describe the divergence operator in terms of the Wiener chaos expansion. Any element u E L2(f~; H) ~ L2(T x f~) (which can be regarded as a square integrable process parametrized by T) has an orthogonal expansion of the form

oo u(t) = ~ In(In(., t)), n=0

(1.20)

where for each n >_ 1, fn E L2(T n+l) is a symmetric function in the first n variables. P r o p o s i t i o n 1.5.2 Let u E L2(T x f~) with the expansion (1.20). Then u belongs to Dom ~ if and only if the series 6(u) = ~ In+10~)

(1.21)

n=0

converges in L2(f~). Equation (1.21) can also be written without symmetrization, because for each n, In+~ (fn) = I~+1 (fn). However, the symmetrization is needed in order to compute the L 2 norm of the stochastic integrals.

Proof: Suppose that G = In(g) is a multiple stochastic integral of order n > 1 where g is symmetric. Then we have the following equalities: E ((u, DG)H) = fT E (In-1 (fn-1 (', t))nIn-1 (g(', t))) #(dr) = n(n - 1)! fT(f~_l(., t), g(., t))L2(T--~)p(dt) = n!(fn-a,g)L2(TN = n!ifn-l,g)L2(TN =

S (In(]n_i)In(g))

= E (In(fn_l)a)

.

Suppose first that u E Dom 5. Then from the above computations and from formula (1.7) we deduce that

E(6(u)O) = E([n(L-1)G) for every multiple stochastic integral G = In(g). This implies that In(fn-a) coincides with the projection of 6(u) on the nth Wiener chaos. Consequently, the series in

137

(1.21) converges in L2(f~) and its sum is equal to 5(u). The converse can be proved by a similar argument. [] For any set A C T we will denote by ~A the a-field generated by the random variables {W(B), B C A, B E Bo}, where B0 = {B e B, #(B) < co}. L e m m a 1.5.1 Let A E B and let F be a random variable in L2(f~, ~-Ac, P) N ]D 1'2. Then D t F = 0 for all (t,w) E A • f~ a.e. with respect to the measure # | P. Proof: Suppose that F = f ( W ( B 1 ) , . . . , W(Bn)), where f E C~(IRn), and Bi C A c, B~ E B0 for all i = 1 , . . . , n. Then we have DtF = 0~(W(B,),...,

W(B~))IB,(t),

which implies that D t F = 0 for all (t, co) E A • fk Finally in the general case it suffices to consider a sequence Fn as above converging to F in L ~. [] The next lemma allow us to interpret the operator 5 as a stochastic integral with respect to the white noise W. L e m m a 1.5.2 Let A E Bo and let F be a random variable in L2(f~,.TA~,P). the process u = F I A belongs to Dom 5 and

Then,

5(F1A) = F W ( A ) . Proof:

Suppose first that F E ID 1'2. Then, using (1.9) and Lemma 1.5.2 we obtain 5(FIA) = F W ( A ) - fT DtF1A(t)#(dt) = F W ( A ) .

The general case follows by a limit argument.

[]

The following lemma provides additional examples of processes in the domain of which are not smooth. L e m m a 1.5.3 Let A E 13o. Let {u(z), x E IRm} be an ~Ac-measurable random field with continuously differentiable paths such that E \lxl_ 0} of contraction operators on L2(f~) defined by TtF = f i e-ntJnF,

(2.1)

r~=0

where Jn denotes the projection on the nth Wiener chaos. This semigroup is called the Ornstein-Uhlenbeck semigroup.

2.1

Mehler's formula

The following result is known as Mehler's formula.

Proposition 2.1.1 Let W ' = {W'(h), h E H } be an independent copy of W . for any t >_ 0 and F C L2(f~) we have TtF = E ' ( F ( e - t W + v/1 - e-2tW')),

Then

(2.2)

where E' denotes the mathematical expectation with respect to W ' . Note that the right-hand side of (2.2) is well defined because { e - t W ( h ) + x/1 - e-27W'(h), h e H } is a centered Gaussian family with the same covariance as W. Proof: Both Tt and the right-hand side of (2.2) give rise to linear contraction 1 2 operators on L2(ft). Thus, it suffices to show (2.2) when F = e x p ( ~ W ( h ) - ~A ),

142

where h E H is an element of norm one and A E IR, We have

E'(exp(e-tkW(h)+x/X-e-2tkW'(h)-~k~)) oo

/

= exp(e-tAW(h)

= ~ e-ntAnH,~(W(h)) !e-2,;) 2 } n=0

-

\

= TtF,

because J ~ F = A~H~(W(h)).

[]

Mehler's formula implies t h a t the operator Tt is nonnegative and t h a t it is a contraction o n / 2 ( F t ) for any p > 1. Moreover the semigroup {Tt, t >_ O} is continuous in/_2(~) for any p _> 1.

2.2

Hypercontractivity

The operators Tt verify a property (cf. Nelson [73], Neveu [74]) called hypercontractivity, which is stronger than the contractivity in L p, and which says t h a t

[ITtFllq(t) p, t > 0 and F E / y ( ~ , )c, p ) . consequence of the hypereontractivity property it can be shown t h a t for any q < oc the norms It' II~ and II" IIq are equivalent on any Wiener chaos 7~n. In t > 0 such t h a t q = 1 + e~t(p - 1). Then for every F E 7fin we have e-"]lFllq = IlT, Yllq 1 the operator J~ is bounded in LY for any 1 < p < oo, and

1)~llFllp II&FIIp 2

if p < 2 .

In fact, suppose first t h a t p > 2, and let t > 0 be such t h a t p - 1 = e st. Using the hypercontractivity p r o p e r t y with the exponents p and 2, we obtain

IIJ, Vll~ = ~'lIv, J~Fllp < e~*lIJ.Fll~ _< e~qIFll2 _< e~'lIFll,.

(2.5)

If p < 2, we use a duality argument. Consider a sequence of real numbers {r linear operator Tr : 5~ -+ P defined by

n k 0}. This sequence determines a

oo

TcF = ~ r

V e P.

n=0

We remark t h a t the operators Tt are of this type, the corresponding sequence being e -'~t. It would be useful to know whether such a multiplication operator is bounded in all Lp, p > 1. The following examples provide sufficient conditions for this property to hold.

143 Examples:

1.

Suppose that r oo -k for n > N and for some ak E ]R such that = ~k=oakn --k < or T h e n the operator Tr is bounded i n / f for any 1 < p < cr

oo ~k=o lakl N

Proof: By duality, and taking into account that Tr is selfadjoint, we can assume oo 7-~ that p _> 2. Moreover it suffices to show that Tr is bounded i n / f on ~n=g ~. Fix M F C ~ = g TI~ with M > N. We have

IIT~FIIp--

akn -k J . F


1. This follows immediately from the equation (1 + n) -c~ = F(ol) -1 e-(n+Utta-ldt,

/7

which implies Tr = P(O~)-1

e-ttc~-lTtdt. ~0 ~ 1 7 6

With the notations of the next section we have Tr = (I - L)-%

(2.s)

144

Given a sequence of real numbers r = {r n >_ 0}, define T~+ = X:~n=0r 1)jn. The following commutativity relationship holds for a multiplication operator T~: D(TcF) = T4)+(DF ),

(2.9)

for any F E 7". In fact, if F belongs to the nth Wiener chaos, we have

D(TcF) = D ( r

= r

= Tr

In particular we have D(TtF) = e tTt(DF), and by iteration we obtain

nk(Tt F) = e-ktTt(nkF)

(2.10)

for any F E 7", and k >_ 1. The following two properties hold: (A) Let F C ]Dk'p. Then TtF also belongs to IDk'p, and lim tl0

IlTtf

FIIk,p = 0.

-

(2.11)

Indeed, from (2.10) it follows that Tt is a contraction operator with respect to any seminorm II-Ifk,p. This implies that TtF E IDk'p. The continuity at the origin with respect to the norm of IDk'p is immeditate. (B) Let F E LP(fl). Then TtF E fTk>~ IDk'p, and

IIDk(TtF)llp < Ck,pt-kllFIIp,

(2.12)

for any k > 1, t > 0, and p > 1. In particular TtF E ID~ for any random variable F which has moments of all orders. Proof: It suffices to show (2.12) for any polynomial random variable. Suppose first that k = 1. We have from Mehler's formula

D(TtF)

= -

D ( E ' ( F ( e - t W + x/'l - e-2tW'))) e-t ,~_...:~_2tE'(D'(F(e-tW + v/1 _ e-2tW'))). X/1 -- e -'2t

We recall that for any random variable G E ID1'2 we have

IIE(DC)II,

= IIJz6'll2 = c,~llJ, Cllp _< 411all,,.

Hence,

/

E (IID(T,F)II~,) 0 (making the changes of variable cos0 = y and y = e - - r ) f0 ~

sin0c~ dO v/lr[ log cos 2 01

1 ~/~

(2.16) + 1)

Suppose that {W'(h), h C H} is an independent copy of the Gaussian family {W(h), h E H}. For any 0 C IR, F C L~ 5c, P) we set

RoF = F ( W cos 0 + W ' sin 0). With these notations we can write the following expression for the operator D ( - C ) -1. L e m m a 2.4.1 For every F C P such that E(F) = 0 we have

:/_} .'(J(RoF)){(0)d0

(2.17)

Proof: Suppose that F = p(W(hl) .... , W(hn)), where h i , . . . , hn C H and p is a polynomial in n variables. For any 0 E ( - 2 , -~) 2 we ]clave ReF = p ( W ( h 0 cos 0 + W'(hl) sin 0 , . . . , W(hn) cos 0 + W'(h,~) sin 0), and therefore

D'(R0P)

-

(W(hl)cos0 + W'(hl) Sin 0, i:l

. . . , W(h,~) cos 0 + W'(h~) sin 0) sin Oh~ = sin ORo(DF).

147

Consequently, using (2.2) we obtain

E'(D'(RoF)) = sin OE'(Ro(DF)) = sin OTt(DF), where t > 0 is such that cos 0 = e -t. This implies

E'(D'(RoF)) = ~ sinO(cosO)~J~(DF). rtmO

Note that since F is a polynomial random variable the above series is actually the sum of a finite number of terms. By (2.16) the right-hand side of (2.17) can be written as

(/-} sinO(cosO)ncP(O)dO)Jn(DF):~ n=0

n~+lJn(DF).

n=O

Finally, applying the commutativity relationship (2.9) to the multiplication operator defined by the sequence r = ~t, n _> 1, r = 0, we get

Tc+DF = DTcF = D(-C)-~F, and the proof of the lemma is complete.

[]

Now with the help of the preceding equation we can show that the operator D C -1 is bounded from LP(ft) into /2(fi; H) for any p > 1. This property will be proved using the boundedness in L p of the Hilbert transform. We recall that the Hilbert transform of a function f E C~(IR) is defined by

H f(x) = f a f ( x + t) --t f ( z - t) dt" The transformation H is bounded i n / 2 ( I R ) for any p > 1 (see Dunford and Schwarz [27], Theorem XI.7.8). Henceforth % and Cp denote generic constants depending only on p, which can be different from one formula to another. P r o p o s i t i o n 2.4.1 Let p > 1. There exists a finite constant cp > 0 such that .for any

F E P with E(F) = 0 we have ]]DC-IFH, < %]]Fl) p. Proof." Using (2.17) we can write E (IIDC-1FII~H) = E = a;~EE '

E'(D'(RoF))~(

( (/: W'

} E'(D'(RoF))~(O)dO

)')

,

where ap = E(I[I p) with [ an N(O, 1) random variable. We recall that for any G E L2(f~',.T',P ') which belongs to the domain of D' the Gaussian random variable W'(E'(D'G)) is equal to the projection J~G of G on the first Wiener chaos. Therefore, we obtain that

E (IIDC-1FllPH) = ap'EE'

J'lRoFqo(O)dO .

148

If g : [ - 2 , ~] --~ 113 is a Lipschitz function with values in some separable Banach space IB the product (z(O)g(O) does not belong to LI([t--~, 2 2];]13) unless g(O) = O. Nevertheless we can define the integral of the product ~g in the following way

?0

~(O)g(O)dO =

/o

~ ~(O)[g(O) - g(-O)]dO.

Notice t h a t d~RoF vanishes at 0 = 0 but this is no longer true for account this remark we can write

RoF. Taking into

E(HDC-11IJHP): o~plFjl~t(J; (/? RoF~(O)dO)p) 1 and any integer k > 1 there exist positive and finite constants Cp,k and Cp,k such that for any F E P,

c,,k

_< (IC Fl')

[E (11 Fll ,o ) + E(IFf')]

(2.20)

Proof: The proof can be done by induction on k. The case k = 1 corresponds to Proposition 2.4.2. In order to illustrate the m e t h o d let us describe the proof of the left-hand side of (2.20) for k = 2.

150 Notice that if {[~, n _> 1} is a family of independent random variables defined on the probability space ([0, 1], B([0, 1]), A) (A is the Lebesgue measure), with distribution N(0,1), and {an, 1 < n < N} is a sequence of real numbers, we have, for each p > 1,

riO,l] N~=an~n(t) Pdt=Ap(n=~la2) ,

(2.21)

x2

where Ap Suppose that F = p(W(hl),..., W(hn)), where the hi's are orthonormal elements of H. We fix a complete orthonormal system {e~, i _> 1} in H which contains the h{s. With these notations, using Eq. (2.21) and Proposition 2.4.2, we can write =

E(IID2FII~|

= E

1 (see example 1). Hence, we obtain

E(]]D2FII~|

dt

H

i=1

C

dtds

= DCRF

< cpE(IIDCRF]]PH) < e'pE(IC2RF[p) ~ c'~E(]C2FIP). []

Khintchine's inequalities also allows us to establish the inequalities (2.20) for polynomial random variable taking values in a separable Hilbert space V. Let us now introduce a continuous family of Sobolev spaces. For any p > 1 and s E IR we will denote by III ' IIIs,pthe seminorm IIIF[I]s,p = ]](I - f)fFIIp,

where F E 7) is a polynomial random variable. Note that ( I - L)~ = ~n~__0(1+n)fJn. These seminorms verify the following properties:

151

(i) IiiFiIkp is increasing in b o t h coordinates s and p. The monotonicity in p is clear and in s follows from the fact t h a t the operators ( I - L) ~ are contractions in /2, for all a < 0, p > 1 (see (2.8)). (ii) The seminorms I]1" []]s,v are compatible, in the sense t h a t for any sequence F~ in P converging to zero in the norm IBm.]Jkp, and being a Cauchy sequence in any other norm aim-]Ni~,,p,they also converge to zero in the norm I1[" ]]l~',p'. We define IDs'p as the completion of P with respect to this norm. Remarks: 1. [HF[[[o,v= ]]F[[0,v = ]]FI[;, and ]13~ = /2(ft). For k = 1 , 2 , . . . the seminorms [[[' Hik,vand [[. [[k,p are equivalent due to Meyer's inequalities. In fact, we have k

k

[iiPlilk,, = ]i(I - L)~FII, _< iE(F)I + ]]R(-L)~F]I;, k

where R = Zo=I

Notice that R - -

r

{ 0i 88

with

n>ln=0

where h(z) = (1 + x)~ in analytic in a neighbourhood of 0. Therefore, the operator R is bounded i n / 2 for all p > 1. Hence,

IllFlllk,p < co(llFllv + [I(-L)~FIIp) < c'p(llFllp + IIDkFIIL,(n;H~)) I! _< c~llFIIk,v. In a similar way one can show the converse inequality. Thus, the Sobolev spaces IDk'v coincide with those defined by means of the derivative operator. 2.

From p r o p e r t y (ii) we have IDs'~ C ]13r

if p' < p and s ~ 0 the operator ( I - L ) - ~ is an isometric isomorphism (in the norm II1"tits,p) b e t w e e n / 2 and IDs'p and between 113-s'p a n d / 2 for all p > 1. As a consequence, the dual of ID~'p is ID -~'q where -1 p +-1q = 1. I f s < 0, the elements o f l D ~'p may not be ordinary random variables and they are interpreted as distributions on the Gaussian space. Set ID - ~ = [Js,v ID~'p. The space 113- ~ is the dual of the space 113~ which is a countably normed space. 4. Suppose t h a t V is a real separable Hilbert space. We can define the Sobolev spaces ID~'v(V) of V-valued functionals as the completion of the class P v of V-valued polynomial r a n d o m variable with respect to the seminorm n[F[[[s,v,v defined in the same way as before. T h e above properties are still true for V-valued functionals. The main application of Meyer's inequalities is the following continuity theorem. P r o p o s i t i o n 2.4.3 Let V be a real separable Hilbert space. For every p > 1 and

s C IR, the operator D is continuous from ID~'P(V) to IDs-~'v(V| and the operator 5 (defined as the adjoint of D) is continuous from IDS'V(V | H) to ID~-LP(V).

152

Proof: We will only proof the continuity of D for V = ]R. The continuity of the adjoint operator follows by duality. For any F E T~ we have (I-

L)~ D F = D R ( I -

L)~ F,

s

where R = ~ = 1 ( h-~ ~ )~ J~. Notice that R = Tr with

{

' o:0o1

where h(x) = ( ~ 1) ~ k in analytic near O. Therefore, the operator R is bounded in L p for all p > 1, and we obtain

I1(I- L)fDFI[p

=

[IDR( I - L)fFIIp 1. Then we have II~(u)ll~ < c,

(llJoullH + IIDulIL~(~;H|

.

(2.22)

Proof." From Proposition 2.4.3 we know that the operator ~ is continuous from ID~'P(H) into LP(f~). This implies that II~(u)llp _< cp

(IlulIL~r

+ IIDulIL,(a;H|

.

On the other hand, we have

IlulIL~ 1. Suppose that SUPn HFnHs,p < oo for some s > O. Then F C ]Ds'p.

Proof:

We know that

sup I1(I

-

L)~F~IIp
liP(1 - ~,,(C))ll~ + . Hence, the inequality

L Fd#'7

(2.24)

holds for any F in the class/2 := { f ( W ( e , ) , . . . , W(en)), n > 1, 0 < f < 1, f Borel}. In order to complete the proof of the lemma and taking into account Daniell's theorem, it suffices to show t h a t given a sequence of random variables {Fro n >

154

1} C s

uniformly bounded by one, such that 0 ___ Fn(w) I 0 for all ~ C ~, then

f~ F,~d#n I 0. We can write /~ Fnd#, 7 < (rl, 1)llFn(1 - ~a(G))ll~r + (7, 99a(G))

(2.25)

Suppose that ~ E ]D -k'p. The second summand in the right-hand side of (2.25) can be estimated as follows: (V, ~a(G)) = ((I - L)-+r/, (I - L)~9~a(G)>
x))l/q,,6 (

D~)

,,p.

In the same way and taking into account the relation E[6(DF/llDFll2H)] = 0 we can deduce the inequality

p(x) < ( P ( r
3.

E ( I D ~ H t l p) +

sup r,s e [0,T]

E

((/0~ [D~,~Htl2dt) w2)

< c~,

158

(ii) IH, I > p > 0 for some constant p. Consider the martingale Mt = f~ HsdWs, and denote by pt(x) the probability density of Mr. Then the following estimate holds

P(IMt[>

pt(x) p-~-a" The constant c depends on A, p, and p.

Proof of (3.7): We will apply Lemma 3.1.1 to the random variable Mr. We claim that Mt C lD 2'2 for each t E [0, T]. In fact, note first that Mt E ]131'2 because the process H belongs to ]D2'2(H), due to condition (i), and the operator ~ is continuous from ]D2'2(H) into ]131'2. Furthermore, using (1.10) we get DsMt = H,l{s y)dy

_ IHt] >_ p > 0 for some constants p and M,

then using the martingale exponential inequality we obtain 1 exp(_ q IzJ ~ 22t ) "

(3.12)

160

3.2

R e g u l a r i t y of d e n s i t i e s a n d c o m p o s i t i o n o f ternp e r e d d i s t r i b u t i o n s w i t h e l e m e n t s of ID- ~

The results obtained in the last section for onedimensional random variables can be extended to the multidimensional case. Furthermore, under additional smoothness and integrability conditions one can show that the probability density of a random vector F is infinitely differentiable. The basic assumptions are introduced in the following definition of nondegenerate random vector. D e f i n i t i o n 3.2.1 We will say that the random vector F = ( F 1 , . . . , F TM) E (lDo*)m is nondegenerate if the matrix 7F is invertible a.s. and

(detTF) - l e

A LP(~) 9

(3.13)

p>l

For a nondegenerate random vector the following integration by parts formula plays a basic role. P r o p o s i t i o n 3.2.1 Let F = (F 1. . . . , F "~) 6 (lD~) 'n be a nondegenerate random vector in the sense of Definition 3.2.1, let G E ]D~ and let g E C~(1Rm). Then (detTF) -1 6 I]9~176 and for any multi-index a 6 { 1 , . . . , m } k, k > 1, there exists an element Ha(F, G) E lDo* such that:

E[(O,g)(F)G] = E[g(F)H~(F, G)].

(3.14)

Moreover the elements Ha(F, G) are recurs@ely given by: m

H(,)(F,G)

E6

:

(G(TF1)i3DFO ,

j=l

Ha(F,G)

H~,(F,H(~ 1...... k_I)(F,G)).

=

Proof: Let us first show that (det 7F) -1 C lDo*. For any N > 1 we have (detTF + I ) -1 C lDo*, because the random variable (det7F + 1 ) - 1 can be written as the composition of det7F with a function in C~r It is not difficult to show using (3.13) that {(det 7F + ~1) -1 , N > 1} is a Cauchy sequence in the norms H' ]]k,p for all k,p, and we obtain the desired result. Also 7/1 E IDO*(IRm• By the chain rule we have rrt

D[g(F)] = ~-~(Oig)(F)DF i, i=1

hence, m

(D[g(F)], DFJ}H = ~"(Oig)(F)7 ij, i=1

and

as a consequence, m

(Oig)(F) = ~-~.(D[g(F)], DFJ)H(TF1) ii. j=l

Finally, taking expectations in the above equality and introducing the adjoint of the operator D we get E[G(Oig) (F)] = E[g(F)g(i)(F, G)],

161

where H(i)(F, G) : ~jm=l ~ (G(~F1)iJDFJ). We complete the proof by a recurrence argument. [] P r o p o s i t i o n 3.2.2 For any p > 1, and for any multi-index c~ there exist a constant

C(p, c~), natural numbers nl, n2, and indices k, d, d', b, b', depending also on p and ct such that ,.y l n]

IIH~(F,G)II~ _ c(p,~) (11 r I1~ IIFII2~IIGII~,,o,) 9 Proof. This estimate is an immediate consequence of the continuity of the operator 5 from IDk+l'p into IDk'p, HSlder's inequality for the II" Ilk,p-norms, and the equality:

D[(~;1) ~j] = _ ~ (~)~k(~l)J~Db~']. k,l=l

[] From Proposition 3.2.1 it follows that the density of a smooth and nondegenerated random vector is infinitely differentiable. We recall that S(IR TM) is the space of infinitely differentiable functions f : IRTM ~ IR such that for any k >_ 1, and for all multi-index/3 E {1,... ,m} j one has sup~ea~ IxlklO~f(x)l < oo. C o r o l l a r y 3.2.1 Let f = ( E l , . . . , F TM) C (ID~176 m be a nondegenerate random vector

in the sense of Definition 3.2.1. Then the density of F belongs to the Schwartz space S(]Rm), and p(x) = E[I{F>x}H(1,2 ....... )(F, 1)], (3.15) where H0,2 ...... )(F, 1) = 5(('TF1DF)mS((',/F1DF) m-1... ~ ( ( ' , / f f l D F ) I ) . - - ) .

Proof." Consider the multi-index a = (1, 2 , . . . , m). From (3.14) we obtain, for any function ~ C C~(IRm), E[(O~O)(F)] = E[{J(F)H~,(F, 1)]. By Pubini's theorem we can write

E[(O~r

=/~m(cq~J)(x)E[l{x 1 and k = 0, 1, 2 , . . . . In particular for every Schwartz distribution T E $'(]Rm), we can define a Wiener distribution T o F C ]D - ~ . Actually

T o F E U N ID -2k,p. k=lp>l

We have t h a t for any fixed point x E IRm the Dirac function belongs to S-2k if k > ~ , and the mapping x --~ 5x is 2j times continuously differentiable from IRm to S - 2 k - ~ . This implies t h a t we can define the "composition" ~x(F) as an element of ID -2k'p for any p > 1 and any integer k such that k > ~ . The density of the random vector F is then given by (6~(F), 1}. Moreover, for any r a n d o m variable G E ID~ and for any x such that p(x) 5r 0, we have

( ~ ( F ) , G) = p(x)E[G[F = x].

3.3

The case of diffusion processes

We can apply the previous results to derive the smoothness of the density for solutions to stochastic differential equations. This provides probabilistic arguments to study heat kernels. Suppose t h a t {W(t), t > 0} is a d-dimensional Brownian motion defined on the canonical probability space f~ = C0(N+; IRa). Let Aj : IR m --* IR,n, j = 0 , . . . , d a system of C ~ functions with bounded derivatives of all orders, and consider the following stochastic Stratonovich differential equation on IR'~: d X t = Y~j=I d A j ( X t ) o dW3t + Ao(Xt)dt, N o = Xo

(3.16)

One can show t h a t Xt E (IDa) m for all t _> 0. The following nondegeneracy condition assures t h a t for each t > 0 the random vector X~ is nondegenerate: (H) There exists an integer k0 _> 0 such that the vector space spanned by the vector fields [AJk, [AJk 1 , [ - " [A31,Ajo]]'.-], 0 < k < k0, where j0 E { 1 , 2 , . . . , d } , j~ E { 0 , 1 , 2 , . . . , d } dimension m.

if 1 < i < k, at point x0 has

In the above hypothesis [A, B] denotes the Lie bracket of the differentiable functions A, B : lR m ~ lRm, defined as

A i OB _ Bi OA ~ . i=i

164

T h e o r e m 3.3.1 Suppose that the above condition (H) is satisfied. Then there exists a positive integer u depending only on ko, and for each p > 1 a positive constant c(p, Xo) such that

H(det ~x,)-lllp _< ct-~,

for all t > O. This theorem is a consequence of the precise estimates obtained by Kusuoka and Stroock in [57].

3.4

Lp e s t i m a t e s o f t h e d e n s i t y a n d a p p l i c a t i o n s

The stochastic calculus of variations can be used to establish Krylov-type estimates which are a useful tool in deriving existence and uniqueness of a solution to stochastic differential equations and to partial stochastic differential equations whose diffusion coefficient is nondegenerate and the drift is not smooth. Let us first establish a preliminary result similar to Lemma 3.1.1. L e m m a 3.4.1 Let a, fl be two positive real numbers such that !~ + ~1 < 1. Let F be an m-dimensional random variable (rn > 1) whose components belong to the space 1D2'~ and E(l(7;l)ii[ ~) < oc, i = 1 . . . . , rn. Then for any nonnegative and measurable function f : IRTM ~ IR we have: E[f(E')] ~

Cc~,B,mHfHmSUiP(E(~r

_~=]]D2F]]L.(f~;H|174

.

Proof: We can assume that the function f is continuous and with compact support. Let r be an approximation of the identity in IRm. We can write E[f(F)]

ej.OJIR"~E [ % ( x -

=

limf


_ O.

Then for each n the random variable Yt~+1 belongs to Np>I IDI'P, and

DJVn+ s*t 1

gj(Y'~,s)+ f t (g'(}~',r),(D~Yn))dWr,

j= l,...,m,

where DY denotes the derivative with respect to the j t h component of the Brownian motion. That is, D~W~ = l{, 1, and using Grownall's lemma we deduce sup sup n

E[IIDW"+lff~]
1. This implies that Yt E flv>l IDZP. Finally we use the chain rule (properly extended to C([0, 1])-valued functions) in order to compute the derivative of g(Y, t), and we obtain:

D~Yt = gj(Y, s) +

2

(g'(V, r), (DgY)>dWr,

j = 1,..., m.

(3.28)

Using Burkholder's inequality we can show (3.24) from the expression (3.28). Similar computations can be done for the second derivative. [] Notice that the inequality (3.20) also holds for a process X such for each t E [0, 1],

Xt is the almost sure limit of a sequence of random variables X~ such that X n solves an equation of the form (3.19) with coefficients g and Fn, where Fn is bounded by K. We can now to derive a convergence criterion for solutions to equation (3.19): C o r o l l a r y 3.4.1 Let hn : IRm x [0, 1] --~ IR be a sequence of measurable functions

uniformly bounded by a constant K1 and converging a.e. to h. Consider a sequence of processes X n solutions of equations Eq(Fn, g), where g satisfies conditions (b) and (c) and Fn is a progressively measurable process bounded by K. Then linmE(fo*lh,~(X:,t)-h(Xt, t)]dt ) = 0 . Proof:

Set

Fix r / > 0. Let r be a nonnegative smooth function with support included in [-1, 1], r = 1 and bounded by 1. Choose R > 0 such that E (/01 ( 1 - r ( ~ ) )

dt) < ~ 9

169

Consider a continuous function 9R,v on IR x [0, 1] bounded by 2KI such that g(x, t) = 0 for all Ix I > R and for a fixed p > m

o

,1]

Ih(x't)-gR'~(x't)lPdx

dt 1 are the exponents appearing in Proposition 3.4.1. Consider the following decomposition

J~=J~+J~+J3+J4n, where

J1n = E ( f o l l h n ( X : , t ) - h ( X ~ , t ) l d t )

J• = E(follh(X,,t)-gR,v(Xt,t)ldt)

J: = Z(follg~,~(x:,t)-g~,.(x.t)ldt) We write

-- E

(/o'r ( ~ :Ih~(Xt, ) ) t ) - h(Xg, t)ldt

+ E(fo 1 (1-~(X--~R))lhn(X2,t)-h(Xg,

t)ldt ) ,

and we make similar decompositions for the terms 3~ and Jn3. In this way we obtain, using Proposition 3.4.1, lira sup J~ n

~("m:~P(/o~(/ "~)'h~(x ~)--~ ~)'~)~0 ~ 1

+limsupE(fol[gR,~(Xt,t)-gR,~(Xt, t)ldt)). Hence, lim sup Jn < Cr/. n

[] C o r o l l a r y 3.4.2 Let gi : C([O, 1];JRm) x [0, 1] --~ ]R, 1 < i < m, be measurable functions satisfying conditions (b) and (c) of Proposition 3.4.1. Consider sequences of

170 progressively measurable processes F, Fn : [0, 1] x ~ ~ ]Rm, and measurable functions f, fn : [0, 1] x ]R --* ]Rm such that TM

limFn(t) n

limf~(t,x)

=

F(t),

dt|

= f(t,x),

a.e.,

dt|

Suppose that for each n > 1 equation Eq(Fm g) admits a solution X ~, and for each t E [0, 1], X~ converges a.s. to a process Xt. Then X solves equation Eq(F,g).

Proof:

It suffices to pass to the limit each term in the equation

x

=x0+ f 0 t F,(s)ds+

rn

[

t

i = l J0

[] Applying the preceding results we can derive existence and uniqueness results for equations of the form Eq(f, g) where f is a measurable and bounded function of Xt and g is a smooth (up to the second order) nondegenerate function. To do this one usually makes use of comparison theorems for stochastic differential equations, and for this reason, one has to consider particular type of equations. In order to illustrate this method we will describe the onedimensional case. P r o p o s i t i o n 3.4.2 Suppose that f : IR x [0, 1] ~ ]R is a measurable function bounded by K and g : IR x [0, 1] ---* IR is twice continuously differentiabIe, with bounded deriva-

tives and such that 0 < cl ~]] = 0,

for all e > 0. The main ingredient of the proof is the following / 2 estimate for the density p~,xof the law of un(t, x). We obtain it using L e m m a 3.1.h sup ~ 1 ~ 0 1 ~

n

n ~ ~o( t , x ) pt,z(r)

drdxdt~()o

for any function qo E C~176(0, 1) 2) with compact support, and for all ~ > 1. B i b l i o g r a p h i c a l n o t e s : For an m-dimensional nondegenerate random vector F, Watanabe has precised the order (s,p) of the negative Sobolev space IDs'p which contains the distribution ~=(F) (cf. [109]). The asymptotic expansion of Xt(eco) using Malliavin calculus has been studied in [108].

Chapter 4 Support theorems In this chapter we apply the stochastic calculus of variations to study the properties of the support of the law of a random vector. We also discuss the characterization of the support of the law using the so-called skeleton approximation, and the application of the Malliavin calculus to the proof of Varadhan-type estimates.

4.1

Properties of the support

Given a random variable F : ~ ~ S with values on a Polish space S, the topological support of the probability distribution of F is defined as the set of points x E S such that P(d(F, x) < ~) > 0 for all c > 0. The connected property of the support of the law of a finite-dimensional random variable vector was established by Fang [28]: P r o p o s i t i o n 4.1.1 Let F E (]DI'P)m for some p > 1. Then the topological support of

the law of F is a closed connected subset of IRm. Proof: Suppose that supp P o F -1 is not connected. There exists two nonempty disjoint closed sets A and B such that supp P o F -1 = A U B. For each integer M >_ 2 let CM : IRm --* IR be an infinitely differentiable function such that 0 < ~2M _~ 1, ~)M(X) = 0 if IXl > M, ~)M(X) 1 if Ixl < M - 1, and SUPx,M IVCM(X)I < ce. Set AM = AN{Ix I ~_ M} and BM = B(3{Ix [ ~_ M}. For M large enough we have AM ~ 0 and BM ~ O, and we can find an infinitely differentiable function fM such that 0 < fM ~_ I, fM = 1 in a neighborhood of AM, and fM = 0 in a neighborhood of =

BM. The sequence (fM~M)(F) converges a.s. and in L p to I{F~A} as M tends to infinity. On the other hand, we have m

D[(fMOM)(F)]

m

= ~(r

i + ~(fMOiOM)(F)DF ~

i=1 m

i=1

= ~-~(IMOiOM)(F)DF i i=1

Hence,

supIID[(fMOM)(F)]IIH M

~

~8up IIO~MII~IIDF~IIH ~ L '~. i=1

M

175

By Lemma 2.4.2 we obtain t h a t l{FcA} belongs to ID I'p and, due to Lemma 1.4.2, this is contradictory because 0 < P(F E A) < 1. [] As a consequence, the support of the law of a random variable F E ]D I'p, p > 1 is a closed interval. The next results provides sufficient conditions for the density of F to be nonzero in the interior of the support. P r o p o s i t i o n 4.1.2 Let F E IDI'p, p > 2, and suppose that F possesses a locally

Lipsehitz density p(x). Let a be a point in the interior of the support of the law of F. Then p(a) > O. Proof:

Suppose p(a) = 0.

Set r = ~p + 2

> 1.

By Lemma 1.4.2 we know t h a t

I{F>~} r ID ~'~ because 0 < P ( F > a) < 1. Fix c > 0 and set

Then qo~(F) converges to l{F>a } in L~(f~), as e $ 0. Moreover, ~ ( F ) C lD 1'~ and 1

D(qo,(F)) = ~-~1[. . . . . +4(F)DF. We have

E(IID(~,(F))II~H) _ 0 for any smooth random variable R because

(~(F), R) = ~o E(r

- F)R),

where ~r is an approximation of the identity. So, from Lemma 2.4.3 there exists a measure r/~ on the ~-field g generated by { W ( e i ) , i _> 1} such t h a t (5~(F), R} = fa Rdrl~ for any g-measurable and smooth random variable R. Notice that p(x) = r/~(ft). Therefore, r/= = 0, which implies that 5=(F) = 0 as an element of ID -~176For any multiindex a we have

Oc~p(x) = Oc~(5~(F), 1) = ((O~5x)(F), 1). Hence, it suffices to show that (O~5=)(F) vanishes. Suppose first t h a t a = (i). We can write m

D(5~(F)) = ~-~(O,~5~)(F)DU, i=1

as elements of ID - ~ , which implies m

(O,6=)(F) = ~-~.(D[5=(F)], DFJ)H("/F1) ji = O, j=l

because D [ ~ ( F ) ] = 0. The general case follows by recurrence.

[]

In the case of a diffusion processes, the characterization of the support of the law is obtained by means of the notion of skeleton. A general notion of skeleton is provided by thc following definition. In the sequel we will assume t h a t {gl N, N _> 1} is a sequence of orthogonal projections on H of finite-dimensional range, which converges strongly to the identity. If { e l , . . . , eN} is an orthonormal basis of the image on H N we set H N w = EN=I W(ei)ei.

177

D e f i n i t i o n 4.1.1 Let F : f~ ~ S be a random variable taking values on a Polish space space (S, d). We will say that a measurable function 9 : H ~ S is a skeleton of F if the following two conditions are satisfied." (i) For all e > 0 we have liNmP { d(O~(yINW), F) > ~} = O. (ii) For all h C H, there exists a sequence of measurable transformations T h : ~2 ~ f~ such that P o (Th) -1 is absolutely continuous with respect to P, and for every e>0 l i m s u p P { d ( F o T h , ~ ( h ) ) < c} > 0 . N

P r o p o s i t i o n 4.1.4 Suppose that ~2 is a skeleton o f F in the sense of Definition 4.1.1. Then the support of the law of F is the closure in S of the set {~(h), h ~ H}.

Proof: Step 1: Let us first show that condition (i) implies the inclusion supp (P o F -1) C q~(H). It suffices to show that P ( F E ~5(H)) = 1, and this follows from

P {d(F,O(H))_ P {d(~(gINW),F) 0, P {d(F, qo(h)) < ~} > O. Since P o (Th) 1 is absolutely continuous with respect to P it suffices to show that P {d(F o


0

for some N _> 1, and this follows from condition (ii).

[]

R e m a r k s : Both parts of the proof are independent, in the sense that Step 1 uses only assumption (i) and Step 2 uses (ii), and we could have taken different skeletons ~1 and eP2 in both assumptions. Also we can replace H by a dense subspace H0 such that IINW takes values in H0, and IINW by a sequence of random variables ~N taking values on H0.

4.2

Strict positivity of the density and skeleton

Under stronger hypotheses, and assuming that F is a finite-dimensional random variable, one can characterize the set of points where the density is strictly positive in terms of the skeleton. The following propositions are devoted to this problem. More precisely we want to establish the equivalence between the following conditions:

178 (a) The density p of a random vector F satisfies p(y) > 0 at some point y E ]Rm. (b) Fix y E ]Rm. There exists an element h E H such that O(h) = y and det'/~(h) > 0, where ~ : H ---* ]RTM is a differentiable mapping and 7~ = ( D~s D~J)H" P r o p o s i t i o n 4.2.1 Let F = ( F 1 , . . . , F m) be a nondegenerate vector in (]D~176 m. Sup-

pose that rp : H ---* IRm is an infinitely differentiable function such that 9 and all its derivatives have polynomial growth (i.e., IID(k)r~(h)ll 0). We will also assume that the following condition holds: (H1) limg~oo r

= F, in the norm I1" IIk,p for all k >_ O,p > 1.

Then for each y e IRm (a) implies (b). Proof: Fix y c lRm. We assume that p(y) = E[6~(F)] > 0. For every M > 1 we consider a function o~M ~ C ~ 1 7 6 s u c h that 0 __< O~M __< 1, C~M(X) = 0 if IXl _< ~,1 and OlM(Z) = 1 if Ixl > ~2. The fact that F is nondegenerate implies that lim 0LM(det'yF) = 1,

M~oo

in the norm I1 IIk,~ for all k, p. Consequently, 0 < E[fy(F)] = lim E[fy(F)~M(det'yF)], M~oo

and we can find a positive integer M such that

E[ey(F)~M(det'Yr)] > 0. Our assumptions imply that h ave

(4.1)

~(IINW) E ]D~176 for every N, and for every k > 1 we

Dk(e(II~vW)) = (IIN)| We have

E[~y(F)C~M(det'yf)] = limooE[6y(q~(IINW))~M (det')'r

(4.2)

in all the norms II" Ilk,p- This convergence follows from hypothesis (H1) and the following integration by parts formulas (see (3.14)):

E[~(F)C~M(det ~'f)] = E[l{y K, and/3K(x) = 1 if Ix[ < K - 1. Then r converges to 1 in lD ~176 and we can find an integer K > 1 such that E [f~(+(IINw))c~M (det')'+(nNw)) •K(IIIINwII2H)] > O.

179 This implies that

p{I~(HNW))-yI<e,

det~(nsw)>_I,

IIIINWll2HO,

(4.3)

for every e > 0. Notice that det 7e(nNw) < det"/r because for every t E ]Rm we have

t' ( D( ~ ( H N W ) ), D( aPJ(IINW ) ) }Htj i,j=l

= ~ t'(IIN[(DO')(H~W)],nN[D('~J)(nNw)])Htj i,j=l m

=

IInN(~t~(Dr


_--~,llnNwll~o.

(4.4)

Finally, from (4.4) we can find a sequence of elements hk E Im(YiN) such that [a2(hk) - Yl < ~1 for every k, Hhk]12H< - K, and det'y~(hk) _> ~1. Bounded and closed sets in the image of 1-IN are compact, so we can select subsequence converging to some element h E H which verifies the desired properties. [] In order to formulate and prove the converse implication (that is, that (b) implies (a)) we need some preliminaries. We will denote by Ba(x) the ball of ]Rm with center x and radius a > 0. The following lemma is a somewhat quantitative version of the classical inverse function theorem. For a function g : BI(0) --* ]Rm which is twice continuous differentiable we will denote by IIglIc~ the norm

Ilgllc2 = zeBx(0) sup {Ig(z)l + Ig'(z)l + Ig"(z)l}. For each/3 > 1 there exist constants cr 6 mapping g : BI(O) ~ ]R verifying g(O) = O,

L e m m a 4.2.1

(0, ~), ~ > 0

(4.5) such that any

TM

Ilgl]c2 e} = 0

l ~ P { l l ( D r ) o T ~ - (Dr

>r

lim supP{(T~Dl,(h),t:,pF) o T h > M }

M~oo N

= 0 =

0

f o r some p > m, and for k = 0, 1, 2, 3. Then (b) implies (a). Proof: We fix a point x E ]Rm and an element h E H such that ~ ( h ) = x and det0'v(h) > 0. Consider the elements of H given by hj = (DrlSJ)(h), j = 1 , . . . ,m. Using these elements we can introduce the random mapping g : IRm + IRTM defined by 9(z) = F ( T z W ) - F, where T z W is defined in (4.6). Notice t h a t the random function g has an infinitely differentiable version and for each multi-index c~ = ( c t i , . . . o~k) E {1, 2 , . . . , m } k its derivatives are given by

(O~g)(z) = ((DkF)(TzW), h~l |

O h~}.~k 9

(4.8)

By Sobolev's inequality, if p > m is the exponent appearing in hypothesis (H2), we have 1 sup Ig(z)l _< ~ Izl_

Izl- 0. If d2R(y) = c~ there is nothing to say. Suppose d~(y) < exp. Let h 9 H be such that (I)(h) = y, detv~(h) > 0, and IIh[[2H< d2R(y)+~?. Let f 9 C~(]l:[m). By Girsanov's theorem

E(f(F~)) = e- 2, E , f ( F ' ( W + ~ ) ) e - - T - ) . Consider a function X 9 C~176

0 < X -< 1, such that x(t) = 0 if t r [-2~, 2~], and

x(t) = 1 if t 9 [-7, r]]. Then, if f > 0, we have E(f(F~)) > e-

~o" E x(eW(h))f(F~(W +

)) .

This implies that

Hence, it suffices to show that

limc21~

)) = 0 .

We have

Then, using hypothesis (ii) the expectation Z (x(~W(h))50( F~(W + ~_)~- + ( h ) ) ) converges, as c tends to zero, to E(5o(Z(h))), and this completes the proof of the proposition. [] The majoration estimate requires large deviation assumptions: P r o p o s i t i o n 4.4.2 Consider a family {F~,0 < ~ < 1} of nondegenerate random

vectors, and a function 9 9 Cp (H; IRTM) such that: (i') sup~c(04 ] ]lF~llk.p < co, for each k > 1 and p > 1. (ii') [I(TF,)-IHk _< e-N(k) for any integer k >_ 1. (iii) The family {F,, 0 < e _< 1} satisfies a large deviation principle with rate function d2(y) = ~(h)=y inf Ilhll~,

y 9 ~.

Then lim sup 2e2 logp~(y) < -d2(y). el0

(4.17)

186

Proof: Fix a point y E ]Itm and consider a function X C C~(Rm), 0 < X -< 1 such that X is equM to one in a neighborhood of y. The density of F~ at point y is given by p,(y) -= E(x(F,)fu(FE)). Using Proposition 3.2.1 we can write

E(x(F,)Sy(F~)) = E (l{F,>~}H(1,2,...,m)(Fc, x(F~))) < E([H(1,2 ...... )(F,, x(F,))[) = E([H(1.2 ...... )(F,, x(F,))[l{f,~suppx})

< (P(F, E suppx))~[Ig(1,%...,m)(F,, x(F,))l]p, where [ + ~ = 1. By Proposition 3.2.2 we know that

IIH(1,=,...,~)(F,, x(F,))II. _< C(,)ll'~:~)llkllF, II.,blix(F,)H.',b', for some constants b, d, b', d'. Thus, hypothesis (ii') implies that lime 2 log IIH(t,2 ...... )(F,, x(F,))IIp = O. el0

Finally, the large deviation principle for F, ensures that for e small enough we have

(P(F, E suppx))~ < e -~-~' (inf~e'uppxd=(y)). [] P r o p o s i t i o n 4.4.3 Consider a family {F~, 0 < e _< 1} of nondegenerate random

vectors, and a function r E C ~ ( H ; ]Rm) such that: (i') sup~e(0.1] IIF~l]k,p < oo, for each k >_ 1 and p > 1. (ii') I]('yF,)-lllk < C-y(k) for any integer k > 1. (iii') The family {(F~, 7F,), 0 < e < 1} satisfies a large deviation principle with rate function A2(y,a) = inf [[h]]~,, (y,a) e ]Rm x ]Rm2. 9 (h)=~,~ (h)=a (iv) limN~o~O(eIIgW) = F,, in the norm I[" IIk.p for all k >_ O, p > 1, where {I[ N, N > 1} is a sequence of orthogonal projections of H of finite-dimensional rangle strongly convergent to the identity. Define d2R(y) = infv(h)=u,det~(h)>o Ilhll2H, y E ]Rm and suppose that ~f :=

inf

~I,(h) = y , d e t ?~ (h) > 0

detT~(h) > 0.

Then lim sup 2~2 logp~(y) < --d2R(y). eJ.0

(4.18)

187

Proof.- The proof is similar to that of Proposition 4.4.2. Consider a function g E 1 Set C~176 0 __ g _< 1, such that g(u) = 1 if lu[ < 88 and g(u) = 0 if [u I > ~7. G~ = g(det 7F~)- With the notations of the proof of Proposition 4.4.2 we have E(x(F~)6y(F~)) = E(G~x(F~)Sy(Fe)) + E((1 - G~)x(F~)Sy(F~)). We have E(G~x(F~)~(F~)) = O, because, otherwise, using condition (i') and applying the method of the proof of Proposition 4.2.1, one can find an element eh E H such that ~(eh) = y and 0 < det ~/r < ~, and this is in contradiction with the definition of 7Proceding as in the Proposition 4.4.2 we obtain E((1 - G~)x(F~)6y(F~)) = E (1F,>yH(,,2 ....... )(F,, (1 - G,)x(F,))) /~(]H(1,2 ...... )(Fr (1 - G,)x(F,))[ )

E([H(1,2,...,m)( F~x( F~)ll F,esuppx,detTF >88
1 and any positive integer k we will denote by ILk'p the space /2([0, 1]; IDk'P). Notice that ]L:'2 = ]DI'2(H), a n d / 2 ( [ 0 , 1]; IDira) C IDI'P(H) for p > 2. P r o p o s i t i o n 5.1.1 Let u be a process in the class IL :'2. Suppose that E f3 [[Dutll~dt < oo f o r somc p > 2. Then the integral process {f~ usdWs, 0 < t < 1} has a continuous version.

191

Proof: We can assume t h a t E ( u t ) = 0 for each t E [0, 1] because the Gaussian process f~ E ( u s ) d W s has a continuous version. Applying the estimate (2.22) we obtain

Set Ar =

I fo~(Dou,)2dOI~.

Fix an exponent 2 < a < 1 + a2, and assume t h a t p is close to 2. Applying Fubini's theorem we can write -~Z~[~ -- ~ - a 2

asdt) 2, the Skorohod integral f~ u~dWs has a continuous version. ~klrthermore, if ?~ E]L I'p, p > 2, we have

E( sup I re[0,1]

/:

usdW~l" ) < co.

The next result will show the existence of a nonzero quadratic variation for the indefinite Skorohod integral. Theorem

1,2

5.1.1 Suppose that u is a process of the space ILloc. Then

cr

n-, t~+, u~dW~ E \Jti i=0

)' /o' ~

u~ds,

(5.6)

in probability, as ]~1 "* O, where ~ runs over all finite partitions {0 = to < tl < " "
1, a n d let q e [1,p]. We will denote by element of Lq([0, 1] x f~) defined by lira

/01

sup

s 1 is a sequence of processes which converges in probability to a random field {Y(0), 0 C ]RTM}for each 0 C ]Rm. Suppose that

E(IY.(O) - Yn(O')l ,) _< e,,/do - 0'1%

(5.23)

for all 101, 10'1 < K, n > 1, K > 0 and for some constants p > 0 and ce > m. Then, for any m-dimensional random variable F one has

l ~ r n ( F ) = Y(F), in probability. Moreover, the convergence is in L v if F is bounded. Proof: Fix K > 0. Replacing F by FK := FI{IFI 0 and consider a random variable F~ wich takes finitely many values and such t h a t IFd _< K and IIF - F~II~ 2 and a > m such that -

u,(x')[p)

s

c,,

rx -

for all Ixl, Ix'I _< K, K > 0, where f~ Ct,Kdt < oc. Moreover f~ E(lut(O)12)dt < OO.

Notice that under the above conditions for each t E [0, 1] the random field {ut(x), x C IRm} possesses a continuous version, and the It6 integral f~ u t ( x ) d W t p o s s e s s a continuous version in (t, x). In fact, for all Ixl, Ix'l _< K, K > 0, we have

m, and Izl, Ix'l _< K,

n-1

1

Q+I

< ~ - f i=0 ti+l - ti Jt~

II[us(X) - ut,(x) - Us(X') + ut~(x')l(w(ti+l) - W(h))Hp~ds

ft--1

_< ep,~ ~

,

[t~+~ - t~189 sup

[E(lu~(x) - u~,(z) - u~(x') + u~,(x

!

)1')]"

s6 [ti,ti+ ll

/=0

1 and suppose that IF] 2, and define Xt = etZt (At, Xo(At) ) . (6.18) Then the process X = {Xt, 0 < t < 1} satisfies l[0,t]aX E Dom5 for all t E [0, 1], X E L2([0, 1] • ~), and X is the unique solution of Eq. (6.16) verifying these conditions. Proof: We will only prove the existence. The uniqueness can be shown using similar

T h e o r e m 6.3.1

arguments. Let us prove first that the process X given by (6.18) satisfies the desired conditions. By Gronwall's lemma and using hypothesis (H.1), we have

IX, I < etetn(IXo(A,)l + L

e;l(rs)ds),

(6.19)

which implies suPte[0.1] E(IX, Iq) < oo, for all 2